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ISBN 978-1-59973-464-4 


VOLUME 1, 2016 


MATHEMATICAL COMBINATORICS 
(INTERNATIONAL BOOK SERIES) 


Edited By Linfan MAO 





EDITED BY 


THE MADIS OF CHINESE ACADEMY OF SCIENCES AND 


ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA 


March, 2016 





Vol.1, 2016 ISBN 978-1-59973-464-4 


MATHEMATICAL COMBINATORICS 
(INTERNATIONAL BOOK SERIES) 
Edited By Linfan MAO 


(www.mathcombin.com) 


Edited By 


The Madis of Chinese Academy of Sciences and 


Academy of Mathematical Combinatorics & Applications, USA 


March, 2016 


Aims and Scope: The Mathematical Combinatorics (International Book Series) is 
a fully refereed international book series with ISBN number on each issue, sponsored by the 
MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 
pages approx. per volume, which publishes original research papers and survey articles in all 
aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, 
non-euclidean geometry and topology and their applications to other sciences. Topics in detail 


to be covered are: 


Smarandache multi-spaces with applications to other sciences, such as those of algebraic 
multi-systems, multi-metric spaces,---, etc.. Smarandache geometries; 

Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and 
map enumeration; Combinatorial designs; Combinatorial enumeration; 

Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential 
Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations 
with Manifold Topology; 

Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- 
natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; 
Mathematical theory on parallel universes; Other applications of Smarandache multi-space and 
combinatorics. 

Generally, papers on mathematics with its applications not including in above topics are 
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Linfan MAO Shaofei Du 

Chinese Academy of Mathematics and System Capital Normal University, P.R.China 
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and 


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Beijing University of Civil Engineering and pojitechnica University of Bucharest 
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li International Journal of Mathematical Combinatorics 


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Peking University, P.R.China 

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Chinese Academy of Mathematics and System 
Science, P.R.China 

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Department of Computer Science 

Georgia State University, Atlanta, USA 


Famous Words: 


There is no royal road to science, and only those who do not dread the fatigu- 
ing climb of gaining its numinous summits. 


By Karl Marx, a German revolutionary . 


Math. Combin. Book Ser. Vol.1(2016), 1-7 


N*C*— Smarandache Curve of 


Bertrand Curves Pair According to Frenet Frame 


Stleyman Senyurt , Abdussamet Caliskan and Unzile Celik 


(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey) 
E-mail: senyurtsuleyman@hotmail.com, abdussamet65@gmail.com, unzile.celik@hotmail.com 


Abstract: In this paper, let (a,a*) be Bertrand curve pair, when the unit Darboux vector 
of the a* curve are taken as the position vectors, the curvature and the torsion of Smaran- 
dache curve are calculated. These values are expressed depending upon the a curve. Besides, 


we illustrate example of our main results. 


Key Words: Bertrand curves pair, Smarandache curves, Frenet invariants, Darboux vec- 


tor. 


AMS(2010): 53A04. 


§1. Introduction 


It is well known that many studies related to the differential geometry of curves have been 
made. Especially, by establishing relations between the Frenet Frames in mutual points of two 
curves several theories have been obtained. The best known of the Bertrand curves discovered 
by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve 
theory. A Bertrand curve is defined as a special curve which shares its principal normals with 
another special curve, called Bertrand mate or Bertrand curve Partner. If a* = a+ AN, 
A = const., then (a,a*) are called Bertrand curves pair. If @ and a* Bertrand curves pair, 
then (T,T*) = cos@ = constant, [9], [10]. The definition of n-dimensional Bertrand curves in 
Lorentzian space is given by comparing a well-known Bertrand pair of curves in n- dimensional 
Euclidean space. It shown that the distance between corresponding of Bertrand pair of curves 
and the angle between the tangent vector fields of these points are constant. Moreover Schell 
and Mannheim theorems are given in the Lorentzian space, [7]. The Bertrand curves are the 
Inclined curve pairs. On the other hand, it gave the notion of Bertrand Representation and 
found that the Bertrand Representation is spherical, [8]. Some characterizations for general 
helices in space forms were given, [11]. 

A regular curve in Minkowski space-time, whose position vector is composed by Frenet 
frame vectors on another regular curve, is called a Smarandache curve [14]. Special Smarandache 
curves have been studied by some authors. Melih Turgut and Stha Yilmaz studied a special 











case of such curves and called it Smarandache T Bz curves in the space Ej ([14]). Ahmad T.Ali 





1Received August 9, 2015, Accepted February 2, 2016. 


2 Siileyman Senyurt , Abdussamet Galigkan and Unzile Celik 


studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret 
invariants of a special case, [1]. Senyurt and Caliskan investigated special Smarandache curves 
in terms of Sabban frame of spherical indicatrix curves and they gave some characterization 


of Smarandache curves, [4]Ozcan Bektag and Salim Yiice studied some special Smarandache 











curves according to Darboux Frame in E®, [3]. Kemal Tasképrii and Murat Tosun studied special 





Smarandache curves according to Sabban frame on S$? ((2]). They defined NC-Smarandache 
curve, then they calculated the curvature and torsion of NB and TNB- Smarandache curves 
together with NC-Smarandache curve, [12]. It studied that the special Smarandache curve in 
terms of Sabban frame of Fixed Pole curve and they gave some characterization of Smarandache 
curves, [12]. When the unit Darboux vector of the partner curve of Mannheim curve were taken 
as the position vectors, the curvature and the torsion of Smarandache curve were calculated. 


These values were expressed depending upon the Mannheim curve, [6]. 


In this paper, special Smarandache curve belonging to a curve such as N*C* drawn by 
Frenet frame are defined and some related results are given. 


§2. Preliminaries 


The Euclidean 3-space E® be inner product given by 


C5 \oa?+a3+23 











where (%1,22,2%3) € E®. Let a: I — E® be a unit speed curve denote by {7,N,B} the 
moving Frenet frame . For an arbitrary curve a € E®, with first and second curvature, « and T 








respectively, the Frenet formulae is given by [9], [10]. 


T’ =kN 
N'’=-«T+7B (2.1) 
B'=-1TN 








Figure 1 Darboux vector 








For any unit speed curve a: I > E?, the vector W is called Darboux vector defined by 





W =7TT +B. (2.2) 


N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 3 


If we consider the normalization of the Darboux, we have 








F K 
j = — 2.3 
sine = Ta 88? = TT ee) 
and 
C =sinyT + cos vB, (2.4) 


where Z(W, B) = . 














Definition 2.1([9]) Let a : I — E® and a* : I — E® be the C?— class differentiable unit 
speed two curves and let {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} be the Frenet frames of 


the curves a and a*, respectively. If the principal normal vector N of the curve a is linearly 














dependent on the principal vector N* of the curve a*, then the pair (a, a*) is said to be Bertrand 


curves pair. 


The relations between the Frenet frames {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} are 
as follows: 
T* =coséT +sin6B 


N*=N (2.5) 
B* =—sindT + cos OB. 
where Z(T,T*) =0 


Theorem 2.2((9], [10]) The distance between corresponding points of the Bertrand curves pair 
ES 











an IE? is constant. 








3. For the curvatures and the 








Theorem 2.3((10]) Let (a,a*) be a Bertrand curves pair in 





torsions of the Bertrand curves pair (a,a*) we have 


— Aw = sin? 0 r = constant 
A(1— AK)’ 
(2.6) 
F sin? 0 
oe 





73 








Theorem 2.4([9]) Let (a,a*) be a Bertrand curves pair in E°. For the curvatures and the 





torsions of the Bertrand curves pair (a,a*) we have 


Ke as = «cos6—7TsinJd, 


(2.7) 


a = «sin@+7cos0. 


By using equation (2.2), we can write Darboux vector belonging to Bertrand mate a*. 


W* =7*T* +B". (2.8) 


4 Stileyman Senyurt , Abdussamet Caligkan and Unzile Celik 


If we consider the normalization of the Darboux vector, we have 
C* = siny*T™* + cosy" B*. (2.9) 


From the equation (2.3) and (2.7), we can write 


* 


T Ksind + 7cosé 








sing” = Sooo = Bin(ve + 9), (2.10) 
|W*| || 
ee. K* &cos@ — 7 sind Soule 458) 
C= oo = ool + 9); 
|W*| || 
where ||W*|| = V«*? + 7*? = ||W|| and Z(W*, B*) = y*. By the using (2.5) and (2.10), 


the final version of the equation (2.9) is as follows: 


C* = sinyT + cos yB. (2.11) 


§3. N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 














Let (a, a*) be a Bertrand curves pair in E? and {T*, N*, B*} be the Frenet frame of the curve 
a* at a*(s). In this case, N*C* - Smarandache curve can be defined by 


i * * 
Y(s) = (NT +0"), (3.1) 


Solving the above equation by substitution of N* and C* from (2.5) and (2.11), we obtain 


sinyT’ + N+cosyB 


ae 32 
v(s) a (3.2) 

The derivative of this equation with respect to s is as follows, 
pee Cee eee Eee (3.3) 


ds J2 


and by substitution, we get 


Ty = Cerone one (3.4) 


V IW? — 20'|WI + 9? 


dsy [WIP -2¢'IWI +e? 
7a | een Sa (3:9) 


In order to determine the first curvature and the principal normal of the curve ~(s), we 


where 


N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 5 


formalize 


V2] (wi cos 6 + w3 sin @)T + woN + (—w} sin @ + w3 cos 0) B] 


Ti,(s) = 5 
: [IW I2 — 2v/IWI] + 9] 


; (3.6) 


where 


wi = (—Kcosd + Tsind + y" cos(y + 4)) (|| WI? — 2y'||WI| + v”) — (— Kcos8 
+rsin# + y’ cos(y + 4) (|WIWIl' — e"|W I] — e'IIWII + v'e") 

w2 = (—||WIl? + o/IWI) (IW? — 29] + 9”) 

w3 = (ksind +7 cos — y' sin(y + 4)) (|| WI? — 2¢" |W] + y’”) — (Ksind 
+7 cos 6 — y'sin(y + 4)) (|WIWIl' — e"IWIl — ¢'IIWII + v'e") 





The first curvature is 


2(w4? + W2 + w37) 


Ky = \|T%, ll, ky = |: 
3 [IW 2 — 26 WI] +e?) 


The principal normal vector field and the binormal vector field are respectively given by 


[(w1 cos @ + w3 sin 0)T + w2N + (—w1 sind + w3 cos 6) B] 
LN See (3.7) 
VW1* + W2* + W3 
wa | — 2x sin 0 cos @ + 7(sin? 6 — cos? 6) + y’ siny|T 
+w1[Ksin@ + 7 cos 6 — vy’ sin(y + @)] N + w2[27 sin 6 cos 6 





+k(sin? 6 — cos? 0) + y! cos y] B 
SS (3.8) 
(IW? — 2¢'|| WI + y’?) wi? + we? + w32) 


The torsion is then given by 


det (yo, py") 


OS TW Ae 
_ ¥2(0n + od + up) 
Ty = Oe + + Pe 


where 
n = (y' cos(y + 0) — KcosO + rsin6)” + (kcos@ — T sin @)||W||? 


—(Kcos 6 — 7 sin A)y'||W|| 
\ = («cos d — 7 8in 9)(y' cos(y + 8) — Kcos8 + rsin6)’ + (—||W||? 
+y'||W|])! — («sin + 7 cos 0)(K sin 8 + 7 cos@ — gy! sin(y + 0)’ 
p = (—K sin 6 — rc0s8)||W ||? + («sin @ + 7 cos A)y"||W || + (Ksind 





Mu 
% 


+7 cos — ¢' sin(y + @)) 


6 Siileyman Senyurt , Abdussamet Galigkan and Unzile Celik 


8 =—(—||WI|/? + ¢'||W|) (ksind + 7 cos — ¢ sin(y + 6) 

0 = —|(¢ cos(y + 8) — 0088 + rsin8) (sin8 + 7.c08 0 — ¢' sin(y +8)’ 
+(' cos(y +8) — 0080 + 7sin8)' (sind + 7.cos6 — ¢ sin(y + 6))| 

= (¢' cos(y + 0) — kcos6 + 7 sin 8) ( — ||W||? + ¢’||W]). 


Example 3.1 Let us consider the unit speed a curve and a®* curve: 
a(s) = —=(-—coss,—sins,s) and a*(s) = tea s, sin 8, s)- 
V2 V2 


The Frenet invariants of the curve, a*(s) are given as following: 


Ts) = at sin s,cos s, 1), N*(s) = (—coss, —sins,0) 
B*(s) = yn eth 1),C*(s) = (0,0,1) 
K*(s) = —s,7T*(s) = 


In terms of definitions, we obtain special Smarandache curve, see Figure 1. 


as as 


* “8 


Figure 2 N*C*-Smarandache Curve 


References 


[1] Ali A. T., Special Smarandache curves in the Euclidean space, International J. Math. Combin., 
2(2010), 30-36. 

[2] Bektag O. and Yiice S., Special Smarandache curves according to Darboux frame in Eu- 
clidean 3-space, Romanian Journal of Mathematics and Computer Science, 3(1)(2013), 
48-59. 


[3] Galigkan A. and Senyurt S., Smarandache curves in terms of Sabban frame of spherical 








10 
11 


12 


13 


14 





N*C*— Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 7 


indicatrix curves, Gen. Math. Notes, 31(2)(2015), 1-15. 

Caligskan A. and Senyurt S., Smarandache curves in terms of Sabban frame of fixed pole 
curve, Boletim da Sociedade parananse de Mathemtica, 34(2)(2016), 53-62. 

Caligskan A. and Senyurt S., N*C*- Smarandache curves of Mannheim curve couple ac- 
cording to Frenet frame, International J.Math. Combin., 1(2015), 1-13. 

Ekmekci N. and Ilarslan K., On Bertrand curves and their characterization, Differential 
Geometry-Dynamical Systems, 3(2)(2001), 17-24. 

Gorgiilii A. and Ozdamar E., A generalizations of the Bertrand curves as general inclined 
curve in E”, Commun. Fac. Sci. Uni. Ankara, Series Al, 35(1986), 53-60. 

Hacisalihoglu H.H., Differential Geometry, Inénii University, Malatya, Mat. No.7, 1983 
Kasap E. and Kuruoglu N., Integral invariants of the pairs of the Bertrand ruled surface, 
Bulletin of Pure and Applied Sciences, 21(2002), 37-44. 

Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006 

Senol A., Ziplar E. and Yayl Y., General helices and Bertrand curves in Riemannian space 
form, Mathematica Aeterna, 2(2)(2012), 155-161. 

Senyurt S. and Sivas S., An application of Smarandache curve, Ordu Univ. J. Sci. Tech., 
3(1)(2013), 46-60. 

Tasképrii K. and Tosun M., Smarandache curves according to Sabban frame on S?, Boletim 
da Sociedade parananse de Mathemtica 3 srie., 32(1)(2014), 51-59, issn-0037-8712. 
Turgut M. and Yilmaz S., Smarandache curves in Minkowski space-time, International 
J.Math.Combin., 3(2008), 51-55. 














Math. Combin. Book Ser. Vol.1(2016), 8-17 


On Dual Curves of Constant Breadth 











According to Dual Bishop Frame in Dual Lorentzian Space D} 





Siiha Yilmaz', Yasin Unliitiirk? and Umit Ziya Savei? 
1. Dokuz Eyliil University, Buca Educational Faculty, 35150, Buca-Izmir, Turkey 


2. Kirklareli University, Department of Mathematics, 39100 Kirklareli, Turkey 





3. Celal Bayar University, Department of Mathematics Education, 45900, Manisa-Turkey 
E-mail: suha.yilmaz@deu.edu.tr, yasinunluturk@klu.edu.tr, ziyasavci@hotmail.com 


Abstract: In this work, dual curves of constant breadth according to Bishop frame are 
defined, and applications of their differential equations are solved for special cases in dual 
Lorentzian space D?. Some characterizations of closed dual curves of constant breadth ac- 
cording to Bishop frame are presented in dual Lorentzian space D? . These characterizations 
are made by obtaining special solutions of differential equations which characterize closed 


dual curves of constant breadth according to Bishop frame in dual Lorentzian space D}. 


Key Words: Dual Lorentzian space, dual curve, dual curves of constant breadth, Bishop 


frame, differential equations. 


AMS(2010): 53A35, 53A40, 53B25. 


§1. Introduction 


Bishop frame is used in engineering. This special frame has been particulary used in the study 
of DNA, and tubular surfaces and made in robot. Most of the literature on canal surfaces 
within the CAGD context has been motivated by the observation that canal surfaces with the 
rational spine curve and rational radius function are rational, and it is therefore natural to ask 
for methods which allow one to construct a rational parameterization of canal surface from its 
spin curve and radius function [8]. The construction of the Bishop frame is due to L. R. Bishop 
in [2]. That is why he defined this frame that curvature may vanish at some points on the 
curve. That is, second derivative of the curve may be zero. In this situation, an alternative 
frame is needed for non continously differentiable curves on which Bishop (parallel transport 
frame) frame is well defined and constructed in Euclidean and its ambient spaces [4, 18]. 
Curves of constant breadth have been studied in pure mathematics, optimization, mechan- 
ical engineering, physics and related directions. Basic properties of curves of constant breadth 
can be explained to someone without having any mathematical background knowledge. The 
existence of non-circular curves of constant breadth in the standard Euclidean plane has been 


known since the time of Euler; e.g., the Reuleaux triangle was presented by Reuleaux to horn- 


lReceived May 22, 2015, Accepted February 4, 2016. 


On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D} 9 


blower, the founder of the compound steam-engine. In recent years, mathematical properties 
of the Reuleaux triangle have led to some very important applications. Since a curve of con- 
stant breadth can be freely rotated in a square always maintaining contact to all four sides of 
the square, a Reuleaux triangle can be used for drilling holes of maximum area into squares. 
Another application is given by the basic single-rotor Wankel engine. Its oval-shaped housing 
surrounds a three-sided rotor similar to a Reuleaux triangle. As the rotor rotates and orbitally 
revolves, each side of the rotor gets closer and farther from the wall of the housing, as also 
described above, in view of drilling holes into squares. A Reuleaux triangle is also used in the 
gear for driving a movie film [12]. 

In the classical theory of curves in differential geometry, curves of constant breadth have a 
long history as a research matter [8, 5, 9]. First it was introduced by Euler in [5]. Then Fujivara 
obtained a problem to determine whether there exist space curves of constant breadth or not, 
and he defined the concept ”breadth” for space curves on a surface of constant breadth [6]. 
Furthermore, Blaschke defined the curve of constant breadth on the sphere [3]. Reuleaux gave 
a method to obtain these kinds of curves and applied the results he had by using his method, in 
kinematics and engineering [14]. Some geometric properties of plane curves of constant breadth 
were given by Kose in [11]. And, in another work of Kése [10], these properties were studied in 











the Euclidean 3-space E®. In Minkowski 3-space as an ambient space, some characterizations 





of timelike curves of constant breadth were given by Yilmaz and Turgut in [17]. Also, Yilmaz 
dealt with dual timelike curves of constant breadth in dual Lorentzian space in [16]. 

Dual numbers were introduced by W. K. Clifford as a tool for his geometrical investigations. 
Then dual numbers and vectors were used on line geometry and kinematics by Eduard Study. 
He devoted a special attention to the representation of oriented lines by dual unit vectors and 
defined the famous mapping: The set of oriented lines in a three-dimensional Euclidean space 
“3 is one to one correspondence with the points of a dual space D® of triples of dual numbers 
7]. 


In this paper, we study dual curves of constant breadth according to Bishop frame in 






































dual Lorentzian space D?. We give some characterizations of dual curves of constant breadth 














according to Bishop frame in D?. Then we characterize these kinds of curves by obtaining 











special solutions of their differential equations in D}. 





§2. Preliminaries 











Let E} be the three-dimensional Minkowski space, that is, the three dimensional real vector 














space E® with the metric 





(dx,dx) = —dx? + dx3 + da2, 











where (21,22, 273) denotes the canonical coordinates in E?. An arbitrary vector x of E? is said 


to be spacelike if (z,x) > 0 or x = 0, timelike if (x,x) < 0 and lightlike or null if (v7, 2) = 0 



































and x #0. A timelike or light-like vector in E? is said to be causal. For x € E? the norm is 
defined by ||z|| = ./|(z,x)|, then the vector x is called a spacelike unit vector if (x,2) = 1 and 


a timelike unit vector if (z,2) = —1. Similarly, a regular curve in E? can locally be spacelike, 




















10 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Save 


timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), 
respectively [13]. 


Dual numbers are given with the set 

















V={F=ax+4+ Ea*;x,2* CE}, 








the symbol € designates the dual unit with the property ¢? = 0 for € 4 0. Dual angle is defined 
as 0 =0 +€6*, where @ is the projected angle between two spears and 6* is the shortest distance 





between them. The set D of dual numbers is commutative ring the the operations + and -. The 
set 
































3=DxDxD={G=+fy';,¢* € E*} 




















is a module over the ring D [15]. 











For any @ =a+€a*, b=b+ €b* € D®, if the Lorentzian inner product of @ and b is defined 





by 
<G,b >=< a,b > +€(< a*,b>+<a,b* >), 











then the dual space D® together with this Lorentzian inner product is called the dual Lorentzian 














space and denoted by D? [1]. For @ 4 0, the norm ||@]| of @ is defined by 


Ill =V<%e >. 


A dual vector @ = w + €w* is called dual spacelike vector if (6,@) > 0 or @ = 0, dual 
timelike vector if (@,@) <0 and dual null (lightlike) vector if (@,@) = 0 for & # 0. Therefore, 
an arbitrary dual curve which is a differential mapping onto D?, can locally be dual spacelike, 

















dual timelike or dual null if its velocity vector is dual spacelike, dual timelike or dual null, 











respectively. Also, for the dual vectors a,b € D3, Lorentzian vector product of these dual 





vectors is defined by 
Gx b=ax b+ E(a* x b+ax b*) 


where a x 6 is the classical cross product according to the signature (+,+,—) [1]. 


The dual arc length of the curve ¢ from t; to t is defined as 
t t t 
s= JP Ollat=fle'Olldtt+e f (t,e)dt=st Es", 
ty ty ty 


where ¢ is a unit tangent vector of y(t). From now on we will take the arc-length s of y(t) as 
the parameter instead of t [9]. 

















Let @: I C E— D} be a dual spacelike curve with the arc-length parameter s. The Bishop 





derivative formula of dual spacelike curve @ is expressed as 


fs kiN RH Res 
Ni oa! -ekT, (1) 
ha bP 


On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 11 





where (7, e) =1, (™,™) =e=H41, (No, No) = —e and ki, ko are Bishop curvatures. Here 


n~ 


ky =R(s)cosh6(s), kh = &(s)sinh@(s) 











Let @: I C E> D} be a dual timelike curve with the arc-length parameter s. The Bishop 











derivative formula of dual spacelike curve @ is expressed as 


T' = kN, + koNo, 


Ni= kf, (2) 
NS= keT, 
where (P,P) a -1, (1, ™) — 1, (No, No) = 1 and ki, ko are Bishop curvatures. Here 
do yo 
T= a and % = ig - ig}. Thus, Bishop curvatures are defined by ((1], [2]) 


n~ n~ n~ 


k= K(s) cosh@(s), ko = &(s) sinh 6(s) 


§3. Main Results 


In this section, we give some characterizations of dual spacelike (timelike) curves of constant 











breadth according to Bishop frame in the dual Lorentzian space D?. First, we give the definition 

















of dual spacelike (timelike) curves of constant breadth in D}. Then we characterize these kinds 











of curves by obtaining special solutions of their differential equations in D}. 





Definition 3.1 Let (C) be a dual spacelike (timelike) curve with position vector @ = ((s) in 














3. If (C) has parallel tangents in opposite directions at corresponding points G(s) and A(sq) 
and the distance between these points is always constant, then (C1) is called a dual spacelike 
(timelike) curve of constant breadth. Moreover, a pair of dual curves (C,) and (C2) for which 
the tangents at the corresponding points @(s) and G(sq), respectively, are parallel and in opposite 
directions, and the distance between these points is always constant are called a dual (timelike) 
curve pair of constant breadth. 


3.1 Dual Spacelike Curves of Constant Breadth According to Dual Bishop Frame 





Let @ = G(s) be a simple closed dual spacelike curve in D?. We consider a dual spacelike curve 











in the class I’ as in [6] having parallel tangents fe y and T, in opposite directions at the opposite 
points ¢ and @ of the curve according to Bishop frame. A simple closed dual spacelike curve 
of constant breadth having parallel tangents in opposite directions at opposite points can be 


12 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Savci 


represented with respect to dual Bishop frame by the equation 
@ = G+ FT +N, + AN2, (3) 


where 7,6 and \ are arbitrary functions of s. Differentiating both sides of (4), we get 





da ds Oy se «a 3 a ee A= te 
——S = (dk 1- + (Fkit+—)M, + (-Fko + —)No. | 
dsq ds (7s ephatenkate LE SA ae ees ae = 2) 
Considering T a ai y by the definition 3.1, we have the following system of equations 
dy dsq 
—_—= k —1-— 
Ts cok t+ edko ae 
dd iz 
We YR1, (5) 
dX ~ 
—= Ykp. 
ds ae 


If we call @ as the angle between the tangent of the curve C' at point @ with a given direction 


do dO 
and taking — fe = TF, — a = T* into account, the equation (5) turns into 
Sa 
d ak, «k 
ol, 541 FO); 
do T 
dé ak 
= 7, (6) 
dé T 
do) 
1 1 
where f(0) = =+ Ges 
Tis) a 
ky is k 
Let K, = +, Ky = = and using the system of ordinary differential equations (6), we have 
ae 


= 
the following dual third order differential equation with respect to ¥ as; 


OF, ie KR, «dk 
OY eRe RDO 4 ae(R dk _p oy 
do? do do do 


fet Se he At ace he P10) 
+e k,d0)—— -¢« Kodé —=_ 
(4 id0) We Cia: a de de 


=0 
We can give the following corollary. 


Corollary 3.1.1 The dual differential equation of third order given in (7) is a characterization 











of the simple closed dual spacelike curve @ according to Bishop frame in D3. 





Since position vector of a simple closed dual spacelike curve can be determined by solution 
of the equation (7), let us investigate solution of the equation (7) in a special case. Let Ki, Ko 


On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 13 


n~ 


and f(0) be constants. Then the equation (7) turns to the following form 


BF * a dy 
— +e¢(K2— k?)— =0. (8) 
aps (Ky 2) 


Solution of equation (8) yields the components 


7 = A+ Bcos(\/K?2 — K26) + Csin(,/ K? — 20) 
ar {A+ Boos /R? — 86) + Gsin(,/R? — yo) Rido @) 
n= if {A+ Beos(/R? — R26) + Gsin(\/R? R30) Rodi. 


Corollary 3.1.2 Position vector of a simple dual spacelike closed curve with constant dual 
curvature and constant dual torsion according to Bishop frame is obtained in terms of the 


values of 4, 5 and X as in the equation (9). 


If the distance between opposite points of @ and @ is constant, then we can write that 





|| — Gl] = -7? + 6? + 9? = constant. (10) 
Differentiating (10) with respect to 0 gives 


dy ~dd  ~dr 
See 


= = 0. (11) 
dd dd do 


By virtue of (6), the differential equation (11) yields 
—~6K1(1 +e) +\Ro(1 —€) + f(0) =0,7 =0. (12) 
There are two cases for the equation (12), we study these cases as follows: 


Casel. If Kk 1 =O and Ks = 0 then we find that the components 5 and \ are constants and 


f(6) =0. 
Hence, Dual spacelike curves of constant breadth according to Bishop frame can be written 
as 
@=G+hT+ LM +13No, (13) 


n~ n~ 


where ¥ = i,6= le A= Is; Licbls are constants. 


n~ 


Case 2. If f() =0, then we have a relation among radii of curvatures as 


1 1 
=-==0. (14) 
E T 


14 


Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Savci 


For this case, the equation (7) turns into 





BF a dy Oka ky. 
oT + (RK? - K2) SL SeCk ile een 
do? do do do 
(15) 
+e(f Kid0)F K.d0)y7—— =0 





The equation (15) is a characterization for the components. However, its general solution of 


has not been found. Due to this, we investigate its solutions in special cases. 


Let us suppose that Kk, =k,= 0, then we rewrite the equation (15) as 


3 
- =r (16) 


By this way, we have the components as follows: 


FH=Q +0460, 
6 = constant, (17) 


x = constant. 


3.2 Dual Timelike Curves of Constant Breadth According to Dual Bishop Frame 











Let ¢ = G(s) be a simple closed dual timelike curve in D3. We consider a dual timelike curve 





in the class I as in [6] having parallel tangents - y and T, in opposite directions at the opposite 


points @ and @ of the curve according to Bishop frame. A simple closed dual timelike curve 


of constant breadth having parallel tangents in opposite directions at opposite points can be 


represented with respect to dual Bishop frame by the equation 


@=G+9T +6N, + XMo, (18) 


where ¥, 6 and X are arbitrary functions of s. Differentiating both sides of (18), we get 





oe oa = (FF 4 Khe +1) + Fk14 Oy, t (Fhe Ea (19) 
Considering T,=—-T y by the Definition 3.1, we have the following system of equations 
Hi  Be fe, — 3h 1, 
2 = -7h, (20) 
2 = —~YFkp. 


If we call @ as the angle between the tangent of the curve C at point @ with a given direction 


On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 15 


I 


n pe 
and taking & Ts = = T* into account, we have (20) as follow; 
8 Sa 


dy ahiy. 265 A 

— —é mS A= os 0 ) 

op = f() 

dé ky 

eee ae (21) 
do if 

B_ _sh 

do ca 


n~ 


where f(0) = 


a 
ype 


a 
Let ky = ae es — = and using the system of ordinary differential equations (21), we 


have the following dual third order differential equation with respect to ¥ as; 











Bae.) tee) ods RG. ie Ke 
i - o (22) 
a~ nak ~ n~ .d?Ko d? f (0) 
—(| K,d0é — — Kodé —— — ——=0 
(f #id0)7 i (f Kod0)¥ ae @ 


We can give the following corollary. 


Corollary 3.2.1 The dual differential equation of third order given in (22) is a characterization 


of the simple closed dual timelike curve @ according to Bishop frame in D3. 


Since position vector of a simple closed dual timelike curve can be determined by solution 
of (22), let us investigate solution of the equation (22) in a special case. Let K,, Ky and £0) 
be constants. Then the equation (22) turns into the following form 

ce anes a5, dy 
oo _ (Kh? 4 K2)2 =0. (23) 
do dé 


Solution of equation (23) yields the components 


F = A+ BelKi+k2)0 + Ce (Kit 526, 
f22f { Gi pee Ce (Kit Kayay Ridd, (24) 
Res { Rie Baer Ce Ki+ kD 0} Rd0 


Corollary 3.2.3 Position vector of a simple dual timelike closed curve with constant dual 
curvature and constant dual torsion according to Bishop frame is obtained in terms of the 
values of 7, 6 and A in the equation (24). 


16 Siiha Yilmaz, Yasin Unliitiirk and Umit Ziya Save 


If the distance between opposite points of ¢ and @ is constant, then we can write that 
|| — Bl] = -7? + 6? + 9? = constant. (25) 


Differentiating (25) with respect to 6 gives 


adi 50 52 _ 9 (26) 
"(6 dg dQ 


By virtue of (21), the differential equation (26) yields 


7f(0) =0. (27) 
There are two cases for the equation (27), we study these cases as follows: 
Case 1. If 7 =0 then we find that the components 6 and X are constants. 


Hence, Dual timelike curves of constant breadth according to Bishop frame can be written 

as 
@=G4hT +N, +13No, (28) 
where ¥ = ,6= Ten = is; i ala are constants. 


n~ 


Case 2. If f(0) =0, then we have a relation among radii of curvatures as 








1 
ae =O. (29) 
For this case, the equation (22) turns into 
BF aT + dk, = dk. 
Cet Ra a eae aay 
do dé dé dé (30) 
ee ~ ~ @K 
—(f Kidd} — (f Kad) ==0 


The equation (30) is a characterization for the components. However, its general solution 
has not been found. Due to this, we investigate its solutions in special cases. 


Let us suppose that Kk, =k, = 0, then we rewrite the equation (30) as 


BF 
if); (31) 
do? 
By this way, we have the components as follows: 
a 6 0 302, 
6 = constant, (32) 


r = constant. 


On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual Lorentzian Space D3 17 


References 


1 





oN D oO 


10 
11 


12 


13 


14 
15 


16 


17 





18 








N.Ayyildiz, A.C. Coken, and A.A. Yiicesan, A characterization of dual Lorentzian spherical 
curves in the dual Lorentzian space, Taiwanese J. Math. 11 (4) (2007), 999-1018. 
L.R.Bishop, There is More Than One Way to Frame a Curve, Amer. Math. Monthly 82 
(1975), 246-251. 

W.Blaschke, Konvexe Bereiche Gegebener Konstanter Breite und Kleinsten Inhalts. Math. 
Ann. 76 (1915), 504-513. 

B.Bukctt and M.K.Karacan, The slant helices according to bishop frame of the spacelike 
curve in Lorentzian space, Jour. of Inter. Math. 12(5) (2009), 691-700. 

L.Euler, De curvis trangularibus, Acta Acad Petropol, (1870), 1870. 

M.Fujivara, On Space Curves of Constant Breadth, Tohoku Math. J. (5) (1963), 179-184. 
H.Guggenheimer, Differential Geometry, McGraw Hill, New York, 1963. 

T.Korpinar, E.Turhan, On characterization of image-canal surfaces in terms of biharmonic 
image-slant helices according to Bishop frame in Heisenberg group Heis?, J. Math. Anal. 
Appl. 382 (1) (2011) 57-65. 

O. Kose, 8. Nizamoglu, and M. Sezer, An explicit characterization of dual spherical curves, 
Doga Turk. J. Math., 12 (3) (1988), 105-113. 

O. Kése, On Space Curves of Constant Breadth, Doga Turk. J. Math. (10) 1 (1986) 11-14. 
O. Kose, Some Properties of Ovals and Curves of Constant Width in a Plane, Doga Turk. 
J. Math. 8 (1984), 119-126. 

H.Martini, Z.Mustafaev, A new construction of curves of constant width, Comp. Aid. 
Geom. Des. 25 (9) (2008) 751-755. 

B.O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 
New York, 1983. 

F.Reuleaux, The Kinematics of Machinery, Dover Publications, New York, 1963. 

G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous spatial 
kinematics, Mech. Math. Theory, 11 (1976), 141-156. 

S.Yilmaz, Timelike dual curves of constant breadth in dual Lorentzian space, IBSU Sci. 
J. 2 (2008) 129- 136. 

S.Yilmaz, M.Turgut, On the time-like curves of constant breadth in Minkowski 3-Space, 
Int. J. Math. Combin., 3 (2008) 34-39. 

S.Yilmaz, Bishop spherical images of a spacelike curve in Minkowski 3-space, Int. Jour. of 
the Phys. Scien. 5(6) (2010) 898-905. 


Math. Combin. Book Ser. Vol.1(2016), 18-26 


On (r,m,k)-Regular Fuzzy Graphs 


N.R.Santhimaheswari 


Department of Mathematics 


G.Venkataswamy Naidu College, Kovilpatti-628502, Tamil Nadu, India 


C.Sekar 
Department of Mathematics 


Aditanar College of Arts and Science, Tiruchendur, Tamil Nadu, India 


E-mail: nrsmaths@yahoo.com, sekar.acasQ@gmail.com 


Abstract: In this paper, (r,m,k)- regular fuzzy graph and totally (r,m,k)- regular fuzzy 
graph are defined and compared through various examples. A necessary and sufficient con- 
dition under which they are equivalent is provided. Also (r,m,k)-regularity on some fuzzy 
graphs whose underlying crisp graph is a cycle is studied with some specific membership 


functions. 


Key Words: Degree of a vertex in fuzzy graph, regular fuzzy graph, total degree, totally 
regular fuzzy graph, dm- degree of a vertex in graph, semiregular graphs, (m, k)-regular fuzzy 


graphs, totally (m,k)-regular fuzzy graphs. 


AMS(2010): 05C12, 03E72, 05072. 


§1. Introduction 


Azriel Rosenfeld introduced fuzzy graphs in 1975 [12]. It has been growing fast and has numer- 
ous applications in various fields. A.Nagoor Gani and K.Radha [11] introduced regular fuzzy 
graphs, total degree and totally regular fuzzy graphs. Alison Northup introduced Semiregular 
graphs that we call it as (2,k)-regular graphs and studied some properties on (2, k)-regular 
graphs [2]. 

N.R.Santhi Maheswari and C. Sekar introduced d2-degree of a vertex in fuzzy graphs, total 
d-degree of a vertex in fuzzy graphs, (2, &)-regular fuzzy graphs and totally (2, k)-regular fuzzy 
graphs [14]. Also they introduced (r, 2, /)-regular fuzzy graphs and totally (r, 2, k)-regular fuzzy 
graphs [15]. 

Also they introduced d,,-degree of a vertex in fuzzy graphs, total d,,-degree of a vertex in 
fuzzy graphs, m-Neighbourly irregular fuzzy graphs and totally m-Neighbourly irregular fuzzy 
graphs [16]. Also, they introduced (m, k)-regular fuzzy graphs and totally (m, k)-regular fuzzy 
graphs [17]. 


1Supported by F.No:4-4/2014-15, MRP- 5648/15 of the University Grant Commission, SERO, Hyderabad. 
2Received April 12, 2015, Accepted November 28, 2015. 


On (r,m, k)-Regular Fuzzy Graphs 19 


These motivate us to introduce (r,m, k)-regular fuzzy graphs and totally (r, m, k)-regular 
fuzzy graphs. We make comparative study between (r,m,k)-regular fuzzy graphs and totally 
(r,m, k)-regular fuzzy graphs. Then we provide a necessary and sufficient condition under which 
they are equivalent. Also (r,m, k)-regularity on fuzzy graphs whose underlying crisp graph is 
a cycle is studied with some specific membership functions. 


§2. Preliminaries 


We present some known definitions and results for ready reference to go through the work 
presented in this paper. 


Definition 2.1({9]) A Fuzzy graph denoted by G : (0,4) on graph G* : (V,E) is a pair 
of functions (a,4) where o : V — [0,1] ts a fuzzy subset of a non empty set V and pu : 
VxV = [0,1] ts a symmetric fuzzy relation on o such that for all u, v in V the relation 
p(u,v) = p(uv) < a(u) Aa(v) is satisfied, where o and p are called membership function. A 
fuzzy graph G is complete if u(u,v) = u(uv) = o(u) A o(v) for all u,v € V, where wv denotes 
the edge between u and v. G* : (V,E) is called the underlying crisp graph of the fuzzy graph 
G: (0, pL). 


Definition 2.2({10]) The strength of connectedness between two vertices u and v is u(u,v) = 
sup{u*(u,v)/k = 1,2,...} where p* (u,v) = sup{u(uuz)Ap(uzus)A-+-Ap(up_iv)/u, ua, U2,°°* 
Up—1,U ts a path connecting u and v of length k}. 


Definition 2.3([11]) Let G: (0,4) be a fuzzy graph. The degree of a verter u is dg(u) = 
> u(uv) for uw € E and (uv) = 0, for uv not in E; this is equivalent to dg(u) = > p(uv). 
ux~vu UveE 


Definition 2.4([11]) Let G: (o,) be a fuzzy graph on G* : (V,E). If d(v) =k for allu € V, 
then G is said to be regular fuzzy graph of degree k. 


Definition 2.5({11]) Let G: (0, ) be a fuzzy graph on G* : (V,E). The total degree of a verter 
u is defined as td(u) = Y> u(u,v)+o(u) = d(u)+a(u), uv € E. If each verter of G has the same 
total degree k, then G is said to be totally regular fuzzy graph of degree k or k-totally regular 
fuzzy graph. 


Definition 2.6({14]) Let G : (o,p) be a fuzzy graph. The dz-degree of a vertex u in G is 
d2(u) = >> p?(u,v), where w?(uv) = sup{u(uui)Au(urv) : u, us, v is the shortest path connecting 
u and v of length 2}. Also, u(uv) = 0, for uv not in E. 

The minimum d2-degree of G is 62(G) = A{do(v):v EV}. 

The maximum dz-degree of G is Ao(G) = V{do(v) :v € V}. 


Definition 2.7([14]) Let G: (o,) be a fuzzy graph on G* : (V,E). If da(v) =k for allv € V, 
then G is said to be (2,k)-regular fuzzy graph. 


Definition 2.8([14]) Let G: (o,) be a fuzzy graph on G* : (V,E). The total d2-degree of a 


20 N.R.Santhimaheswari and C.Sekar 


verter u € V is defined as tdo(u) = D> pw? (u,v) +0(u) = do(u) + 0(u). 
The minimum tdz-degree of G is tdo(G) = A{tdo(v) :v € V}. 
The maximum tdz-degree of G is the(G) = V{td2(v) :v EV}. 


Definition 2.9([14]) If each vertex of G has the same total dz - degree k, then G is said to be 
totally (2, k)-regular fuzzy graph. 


Definition 2.10((15]) Jf each verter of G has the same degree r and same d2-degree k, then G 
is said to be (r,2,k)-regular fuzzy graph. 


Definition 2.11((15]) If each vertex of G has the same total degree r and same total d2-degree 
k, then G is said to be totally (r,2,k)-regular fuzzy graph. 


Definition 2.12([16]) Let G: (0,4) be a fuzzy graph on G* : (V,E).. The d-degree of a 
vertex u in G is dm(u) = > (uv), where p™ (uv) = sup{u(uur) A w(uru2) A..., W(Um—10) : 
U, U1, U2,--+;Um—1,U ts the shortest path connecting u and v of length m}. Also, w(uv) = 0, for 
uv not in EB. 

The minimum d»,-degree of G is dm(G) = A{dm(v) :u € Vt. 

The maximum d»-degree of G is Am(G) = V{dm(v) :u € V}. 


Definition 2.13([16]) Let G: (a,) be a fuzzy graph on G* : (V, E). The total dm-degree of a 
vertex u € V is defined as tdm(u) = >> w™ (uv) + o(u) = dm(u) + a(u). 

The minimum tdm-degree of G is tiym(G) = A{tdm(v) :v € V}. 

The maximum tdy-degree of G is tAn(G) = V{tdm(v) :u € V. 


Definition 2.14([17]) Let G: (0, 4) be a fuzzy graph on G* : (V,E). Ifdm(v) =k for allv € V, 
then G is said to be (m,k)-regular fuzzy graph. 


Definition 2.15({17]) If each vertex of G has the same total dm - degree k, then G is said to 
be totally (m, k)-regular fuzzy graph. 


§3. (r,m,k)-Regular Fuzzy Graphs 


In this section, we define (r, m, &)-Regular Fuzzy Graphs and illustrates this with (r, 3, k)-regular 
graph. 


Definition 3.1 Let G: (a,y) be a fuzzy graph on G* : (V,E). If dv) =r and d,,(v) =k, for 
allu € V, then G is said to be (r,m,k)-regular fuzzy graph. That is, if each vertex of G has the 


same degree r and same dy-degree k, then G is said to be (r,m,k)-regular fuzzy graph. 


Example 3.2 Consider G’ : (V,E), where V = {u1, uo, U3, Ua, Us, Us, U7, Ug, Ug, Uo} and 

E = {uyue, ugus, ugua, UsUs, U5 U6, UBUT, U7Ug, UgUg, Ugt10, UioU1}. Define G: (a, w) by o(ui1) = 

0.3, o(u2) = 0.4, o(ug) = 0.5, o(us) = 0.6, o(us) = 0.7, o(ug) = 0.6,0(u7) = 0.5, o(ug) 

0.4, 0(ug) = 0.3, o(uio) = 0.2 and pu(uru2) = 0.3, p(ugus) = 0.4, u(ugus) = 0.3, p(usus) = 
) 


I 


I 


0.4, u(usug) = 0.3, o(ugu7) = 0.4, o(uzug) = 0.3, o(ugug) = 0.4, o(ugui0) = 0.3, o(ui9u1 


On (r,m, k)-Regular Fuzzy Graphs 21 


0.4. 


(ur) = {0.3A0.4A 0.3} + {0.3A0.4A 0.3} =0.3+0.3 = 0.6. 
(uz) = {0.3A0.3A 0.4} + {0.4A0.3A 0.4} =0.3+0.3 = 0.6. 
(uz) = {0.4A0.3A 0.3} + {0.3A0.4A 0.3} = 0.3+0.3 =0.6. 
(ug) = {0.3A.0.4A 0.3} + {0.4A 0.3A 0.4} = 0.3+0.3 = 0.6. 
d3(us) = {0.3A 0.4A 0.3} + {0.4A 0.3A 0.4} = 0.3 + 0.3 = 0.6. 
(us) = {0.4A0.3A 0.4} + {0.3A0.4A 0.3} = 0.3+0.3 = 0.6. 
(uz) = {0.3A0.4A 0.3} + {0.4A0.3A 0.4} = 0.3+0.3 = 0.6. 
(ug) = {0.4A0.3A 0.3} + {0.3A 0.4A 0.3} = 0.3+0.3 = 0.6. 
(ug) = {0.3A.0.3A 0.4} + {0.4A0.3A 0.4} = 0.3+0.3 =0.6. 
d(u;) = {0.3 + 0.4} = 0.7 for i = 1,2,3,4,5,6,7,8,9,10. 








It is noted that, each vertex has the same d3-degree 0.6 and each vertex has the same 
degree 0.7. Hence G is (0.7, 3, 0.6)-regular fuzzy graph. 


Example 3.3 Consider G* : (V, EF), where V = {u,v,w,2,y, 2} and 
E = {uv, vw, wz, ry, yz, zu}. 


u(0.4) 





x(0.4) 
Figure 1 
d3(u) = Sup{0.3 A 0.3 A 0.3,0.3 A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3 
d3(v) = Sup{0.3 A 0.3 A 0.3,0.3A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3 
d3(w) = Sup{0.3 A 0.3 A 0.3, 0.3 A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3 
d3(x) = Sup{0.3 A 0.3 A 0.3,0.3 A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3 
d3(y) = Sup{0.3 A 0.3 A 0.3,0.3A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3. 
d3(z) = Sup{0.3 A 0.3 A 0.3,0.3 A 0.3 A 0.3} = Sup{0.3, 0.3} = 0.3. 


In Figure 1, d(w) = 0.3+0.3 = 0.6, d(v) = 0.6, d(w) = 0.6, d(x) = 0.6, d(y) = 0.6, d(z) = 
0.6. Each vertex has the same d3-degree 0.3 and each vertex has the same degree 0.3. Hence G 
is a (0.6, 3, 0.3)-regular fuzzy graph. 


Example 3.4 Non regular fuzzy graphs which is (m, k)-regular 


22 N.R.Santhimaheswari and C.Sekar 


1. Let G: (o,) be a fuzzy graph such that G* : (V, F), a path on 2m vertices. Let all 
the edges of G have the same membership value c. Then, for i = 1,2,3,4,5,---,m, 


dm (vi) = {uex) A wleigi) A {u(eiz2) +++ A Hlem—144)} 
={cAcAc:::Ach=c. 
dm(Um+i) = {u(ex) A wleit1) } + {u(ei+2) +++ A w(em—144) } 
={cAcAc:::Ach =e. 
dm(v) = c, for all v € V. 


Hence G : (0, 4) is (m,c)-regular fuzzy graph. 
Pore 9,3, 4505+ 2m —1, 
d(vi) = {u(ei-1) + w(ei) = 2c. 
dvr) = {u(e1)} = ©. 


d(vam) = ule2m—1) = 


d(v1) £4 d(u;)  d(vam) for i = 2,4,5,--- ,2m— 1. Hence G is non- regular fuzzy graph which 


is (m, c)-regular. 


Example 3.5 Let G: (o,) be a fuzzy graph on G* : (V, £), a cycle of length > 2m +1. Let 


C1 if i is odd 
Hei) = pees 
membership value x > c, if 7 is even, where zx is not constant and 
dm(v) = min{c,z} + min{z,a} =a +c. = 2c) 
for all v € V. 


Case 1. Let G: (¢,) be a fuzzy graph on G* : (V, E) an even cycle of length < 2m-+2. Then 
d(v;) = «+c, fori = 1,2,4,5,---,2m+1. So, G: (o,) is non-regular (m, k)-regular fuzzy 
graph, since x is not constant. 


Case 2 Let G: (o, 4) bea fuzzy graph on G® : (V, FE) a odd cycle of length < 2m+1. Hence G : 
(a, ) is (m, 2c, )-regular fuzzy graph and d(v1) = 2c1, d(v;) = a#+c, for i = 2,4,5,--- ,2m+1. 


So, G: (a, 4) is non-regular (m, k)-regular fuzzy graph since x is not constant. 


§4. Totally (r,m,k)-Regular Fuzzy Graphs 


In this section, we introduce totally (r, m, k)-regular fuzzy graph and the necessary and sufficient 
condition under which (r,m, k)-regular fuzzy graph and totally (r,m, k)-regular fuzzy graph are 


equivalent is provided. 


Definition 4.1 Jf each verter of G has the same total degree r and same total d»,-degree k, 
then G is said to be totally (r,m,k)-regular fuzzy graph. 


On (r,m, k)-Regular Fuzzy Graphs 23 


From Figure 1, it is noted that each vertex has the same total d3-degree 0.7. 














td3(u) = d3(u) + o(u) = 0.34 0.4 = 0.7 
td3(v) = d3(v) +o(v) = 0.34 0.4 = 0.7 
td3(w) = d3(w) + o(w) = 0.34 0.4 = 0.7 
td3(a) = d3(x) + o(x) = 0.3+0.4 = 0.7 
tds(y) = d3(y) + o(y) = 0.34 0.4 = 0.7 
td3(z) = d3(z) + o(z) = 0.34 0.4 = 0.7 
td(u) = d(u) + o(u) = 0.8+0.4 = 1.2 
td(v) = d(v) +o(v) = 0.84+0.4 = 1.2 
td(w) = d(w) + o(w) = 0.8+0.4 = 1.2 
td(x) = d(x) + o(a) =0.8+0.4 = 1.2 
td(y) = d(y) + o(y) =0.84+0.4 = 1.2 
td(z) = d(z) +o(z) =0.8+0.4=1.2 





In Figure 1, Each vertex has the same total d3-degree 0.7 and each vertex has the same 
total degree 1.2. Hence G : (o, 2) is totally (1.2,3,0.7)-regular fuzzy graph. 


Theorem 4.2 Let G: (o,) be a fuzzy graph on G* : (V,E). Then o is constant function iff 
the following conditions are equivalent: 

(1) G: (0, p) is (r,m,k)-regular fuzzy graph; 

(2) G: (a, p1) ts totally (r,m, k)-regular fuzzy graph. 


Proof Suppose that o is constant function. Let o(u) = c, constant for all wu ¢ V. Assume 
that G : (0, 1) is (r,m, k)-regular fuzzy graph. Then d(u) =r and d»(u) = k, for all u € V. So 


td(u) = d(u) + o(u) and tdy,(u) = dm(u) + o(u) for all u € V. 
=> td(u) =r+cand td,(u) =k +c for allue V. 


Hence G: (0, ) is totally (r +c,m,k+c)- regular fuzzy graph. Thus (1) > (2) is proved. 
Now suppose G is totally (r,m, k)-regular fuzzy graph. 


=> tdy(u) =k and td(u) =r for all ue V. 
=> dm(u) 
=> dm(u) 
=> dm(u) =k—cand d(u) =r—c for allue V. 


+ o(u) =k and d(u) + o(u) =r for allue V. 





+c=k and d(u) +o(u) =r for allu€ V. 


Hence G: (0, 4) is (r — c,m,k — c)-regular fuzzy graph and (1) and (2) are equivalent. 
Conversely assume that (1) and (2) are equivalent. Suppose o is not constant function. 
Then o(u) 4 o(w), for at least one pair u,w € V. Let G: (o,) be a (r,m,k)-regular fuzzy 


24 N.R.Santhimaheswari and C.Sekar 


graph. Then, d»(u) = = (w) = k and d(u) = d(w) = r. So, tdm(u) = dm(u) + o(u) = 
k + o(u) and tdy(w) = dn(w) + o(w) = k + o(w) and td(u) = d(u) + o(u) = r+ a(u) 
and td(w) = d(w) + o(w) = r+o(w). Since o(u) 4 o(w) > k+a(u) 4 k + 0(w) and 
r+o(u) A r+o(w => tdp(u) A tdyn(w) and td(u) 4 td(w). So G : (0,4) is not totally 
(r,m, k)-regular fuzzy graph which is contradiction to our assumption. Let G : (o,~) be a 
totally (r,m, k)-regular fuzzy graph. Then, td,,(w) = tdm(w) and td(u) = td(w). 








=> dm(u) +a(u) = dm(w) + o(w) and d(u) + o(u) = d(w) + o(w) 
= dm(u) — dm(w) = o(w) — Ay lama d(w) 
=o(w)—a(u)# 
dm(u) # dm(w) and d(u) a 


So G: (a, p) is not (r, m, k)-regular fuzzy graph which is a contradiction to our assumption. 











Hence a is constant function. 





Theorem 4.3 If a fuzzy graph G : (0, p) is both (r,m,k)-regular and totally (r,m, k)-regular 


then o is constant function. 


Proof Let G be (r1,m,k1)-regular and totally (r2,m,k2)-regular fuzzy graph. Then 
dm(u) = ky and td,(u) = ko,d(u) = ry and td(u) = re, for all u € V. Now, td,,(u) = ke 
and td(u) = rg, for all u € V. 


=> dm(u) + o(u) = kg and d(u) + o(u) = re for all u € V. 
=> ky + o0(u) = kp and r; + o(u) = 12 for allue€ V. 


=> o(u) = ko — ky and o(u) =r2—1; for allue V. 











Hence a is constant function. 





§5. (r,m,k)- Regular Fuzzy Graph on a Cycle with 
Some Specific Membership Function. 


In this section, (r,m, k)-regularity on a cycle Com, Com+i is studied with some specific mem- 
bership functions. 


Theorem 5.1 For any m > 1, let G: (0,4) be a fuzzy graph on G* : (V,E), a cycle of 
length > 2m. If ws is constant function, then G : (0,4) is (r,m,k)-regular fuzzy graph, where 
r = 2p(uv) and k = p(uv). 


Proof If 4 is constant function say p(uv) = c, then d,,(v) = Sup{(cAc---Ac)), (cAc---Ac} 
= ¢, for all v € V and d(v) =c+c = 2c. Hence G is (2c, m, c)-regular fuzzy graph. 














Remark 5.2 Converse of the above Theorem need not be true. 


Theorem 5.3 For anym > 1, let G: (0,4) be a fuzzy graph on G* : (V,E), a cycle of length 


On (r,m, k)-Regular Fuzzy Graphs 25 
> 2m+1. If u is constant function, then G is (r,m,k)-regular fuzzy graph, where r = 21(uv) 
and k = 2u(wv). 


Proof If u is constant function say u(uv) = c, then dm(v) = {cAc::-Ack+{cAc:--Ack = 
c+c= 2c, for all v € V and d(v) =c+c= 2c. Hence G is (2c, m, 2c)-regular fuzzy graph. 














Remark 5.4 Converse of the above Theorem need not be true. 


Theorem 5.5 For any m > 1, let G: (0,4) be a fuzzy graph on G* : (V,E), an even cycle 
of length > 2m + 2. If the alternate edges have the same membership values, then G : (a, 4) is 
(r,m, k)-regular fuzzy graph. 


Proof If the alternate edges have the same membership values, then 


C1, if 7 is odd 
u(ei) = a2. 
C2, if 7 is even. 


If cy = ce, then p is constant function. So, G: (0, ) is (2c1, m, 2c1)-regular fuzzy graph. 
If cy < ca, then dm(v) = {ar Ncn...cr NCa} t+ {aANe...aANc}=a+a = 2c, for allueV 
and d(v) = c, + cg. Hence G: (a, 1) is (cr + co, m, 2c1)-regular fuzzy graph. 

If cy > cg, then dy(v) = {c1 Aco...c1 Nco} + {er Aco...c1 Aca} = C2 + C2 = 2c, for all 
v € V and d(v) =c, + cg. Hence G: (0, ) is (c1 + co, m, 2cz)-regular fuzzy graph. 














Remark 5.6 Even if the alternate edges of a fuzzy graph whose underlying graph is an even 
cycle of length > 2m + 2 have the same membership values, then G': (0, 4) need not be totally 
(r,m, k)-regular fuzzy graph, since if o is not constant function then G : (a, 4) is not totally 
(r,m, k)-regular fuzzy graph, for any m > 1. 


Theorem 5.7 For any m > 1, let G: (0, 4) be a fuzzy graph on G* : (V, E), a cycle of length 
>22m+1. Let 
C1, if i is odd 


Lei) = one 
C2 > C1, if 1 as even, 


then G: (a, 1) is a (m,k)-regular fuzzy graph. 
Proof Let 
C1, if 7 is odd 


C2 > C1, if 7 is even 


Case 1. Let G: (¢,) be a fuzzy graph on G* : (V, £) an even cycle of length < 2m-+2. Then 
by theorem 6.3, G is (cy + ¢2,m, 2c1)-regular fuzzy graph. 


Case 2. Let G: (o,y) be a fuzzy graph on G* : (V, E£) an odd cycle of length < 2m +1. For 
any m > 1, dm(v) = 2c1, for allv € V. But d(v1) = a +c, = 2c; and d(u;) = ci + ce, for 
i #1. Hence G is not (r,m, k)-regular fuzzy graph. 














26 


N.R.Santhimaheswari and C.Sekar 


Remark 5.8 Let G: (¢,) be a fuzzy graph on G* : (V, £), an even cycle of length > 2m +1. 
Even if 


C1, if 7 is odd 
u(ex) = es. 
c2> Cc, if 7 is even, 


then G need not be totally (r,m, k)-regular fuzzy graph, since if o is not constant function then 


G is not totally (r, m, k)-regular fuzzy graph. 


References 


1 





10 


11 


12 


13 








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of Mathematical Archive, 7(1), 2016, 1-7. 


Math.Combin. Book Ser. Vol.1(2016), 27-83 


Super Edge-Antimagic Labeling of Subdivided Star Trees 


A.Raheem and A.Q.Baig 


Department of Mathematics 


COMSATS Institute of information Technology, Islamabad, Pakistan 
E-mail: rahimciit7@gmail, makbiyik@yildiz.edu.tr, aqbaig1@gmail.com 


Abstract: Let G be a graph with V(G) and E(G) as the vertex set and the edge set 
respectively. An (a,d)-edge-antimagic total labeling of a graph G is a bijection \ from 
the set V(G) U E(G) — {1,2,3,---,|V(G| + |E(G)|} such that the set of edge-weights 
{A(z) + A(zy) + A(y) : ey € E(G)} is equal to {a,a+d,a+ 2d,--- ,a+(|E(G)| — 1)d} where 
the integers a > 0 and d > 0. An (a,d)-edge-antimagic total labeling of a graph G is called 
super (a,d)-EAT labeling if the smallest possible labels are assigned to the vertices of the 
graph G. 


Key Words: Labeling, super (a, d)-EAT labeling, subdivision of star trees. 


AMS(2010): 05C78. 


§1. Introduction 


All graphs in this paper are finite, undirected and simple. For a graph G we denote the vertex-set 
and edge-set by V(G) and E(G), respectively. A (uv, e)-graph G is a graph such that v = |V(G)| 
and e = |E(G)|. A general reference for graph-theoretic ideas can be seen in [24]. In the present 
paper the domain will be the set of all the elements of a graph G and such a labeling is called a 
total labeling. The more details on antimagic total labeling can be seen in [14, 9]. The subject 
of edge-magic total labeling of graphs has its origin in the works of Kotzig and Rosa [17, 18] 
on what they called magic valuations of graphs. The definition of (a, d)-edge-antimagic total 
labeling was introduced by Simanjuntak, Bertault and Miller in [21] as a natural extension of 
edge-magic labeling defined by Kotzig and Rosa. 


Conjecture 1.1([11]) Every tree admits a super edge-magic total labeling. 


In the support of this conjecture, many authors have considered super edge-magic total 
labeling for many particular classes of trees for example[23, 1, 20, 2, 22, 310, 15, 16, 12, 13, 
21]. Lee and Shah [19] verified this conjecture by a computer search for trees with at most 17 
vertices. However, this conjecture is still as an open problem. 

A star is a particular type of tree graph and many authors have proved the magicness 
for subdivided stars. Ngurah et. al. [20] proved that T(m,n,k) is also super edge-magic if 


lReceived April 17, 2015, Accepted December 7, 2015. 


28 A.Raheem and A.Q.Baig 


k =n+3orn+4. In [23], Salman et. al. found the super edge-magic total labeling of a 
subdivision of a star S7” for m = 1,2. Javaid et. al. [16] proved super edge-magic total labeling 
on subdivided star Ky,4 and w-trees. 

However, super (a, d)-edge-antimagic total labeling of GY T(n1,n2,ng,--- ,n,) for differ- 
ent {nj :1<i <r} is still open. 


Definition 1.1 A graph G is called (a, d)-edge-antimagic total ((a,d) — EAT) if there exist 
integers a > 0, d> 0 and a bijection 


A: V(G)U E(G) = {1,2,3,---,v+e} 


such that W = {w(ry) : cy € E(G)} forms an arithmetic sequence starting from a with the 
common difference d, where w(ay) = A(x) + A(y) +A(xy) for every cy € E(G). W is called the 
set of edge-weights of the graph G. 


Definition 1.2 A (a,d)-edge-antimagic total labeling X is called super (a, d)-edge-antimagic 
total labeling if AV(G)) = {1, 2,3,---,v}. 


Definition 1.3. For n; > 1 andr > 3, let G = T(n1, ne, nz,-+- ,n,) be a graph obtained by 
inserting n, — 1 vertices to each of the i-th edge of the star Ky, where 1 <i<r. 


The notion of a dual labeling has been introduced by Kotzig and Rosa [17]. According to 
him, if f is an (a,0)-EAT labeling with magic constant a then f; is also an (a,0)-EAT labeling 
with magic constant a, = 3(v+-e+1)-—a. The following is defined as fi(x) =vu+e+1-— f(z) 
for alla € V(G)U E(G). 


Lemma 1.1{12] If f is a super edge-magic total labeling of G with the magic constant c, then 
the function fi : V(G) U E(G) = {1,2,3,---,u+e} defined by 


v+1-— f(a), for x € V(G), 
filx) = 
2Qv+e+1—f(x), for «€ E(G). 
is also a super edge-magic total labeling of G with the magic constant cy = 4u+e+3-—c. 
We consider the following proposition which we will use frequently in the main results. 
Proposition 1.1([8]) Jf a (v,e)-graph G has a (s,d)-EAV labeling then 


(1) G has a super (s +v+1,d+4+1)-EAT labeling; 
(2) G has a super (s + v+e,d—1)-EAT labeling. 


§2. Super (a,d)-EAT Labeling of Subdivided Stars 


In this section we deal with the main results related to the super (a,d)-EAT labelings. on 
generalized families of subdivided stars for all possible values of d. 


Super Edge-Antimagic Labeling of Subdivided Star Trees 29 


Theorem 2.1 Forn>1 andr >4,G2T(n4+1,n+ 2,2n + 4,14,--+ ,n,) admits a super 
(a,0)-EAT labeling with a = 2v+s5—1 and a super (a,2)-EAT labeling witha =v+s+1, 


where v =|V(G)| and s = (n+5)+ D> [2™-4(n +2)] and nm = 2™-7(n 4+ 2) ford<m<r. 
m=4 





Proof The vertices and the edges of the graph G are v = (2n+4)+ > [2™-3(n + 2)] and 
m=4 
e =v—1. Define the vertex labeling A: V(G) — {1,2,--- , uv} as follows: 


Let A(c) = 1. For even 1 <1; < nj, where i= 1,2,3 and4<i<r: 


1+5, for u=s}, 

Au) = 4 (n+3)-4, for u=2? 
l 

(Q2n+5)—%., for u=a}. 


Nai!) = (2n +5) + S- (2-3 (mn + 2)) — respectively. 


m=4 


For odd 1 <1; <n; and a= (2n+5)+ Y> [2-3 (n + 2)], where i = 1,2,3 and 4 <i<r: 


m=4 











a+ 42, for u=c!, 
A(u) = (a+n-+3) fat for u= x, 
(a+2n+5)— 844, for u= xP. 





and X(ai*) = (a+ 2n+5)+ D> [2™-3(n + 2)] — 42 respectively. 


m=4 





The set of all edge-sums {A(x) + A(y) : ey € E(G)} generated by the above formulas forms 
an integer sequence (a + 1) + 1,(a+1)4+2,---,(a+1)+e, where s = a+2. Therefore, 





by Proposition 1.1, \ can be extended to a super (a,0)-EAT labeling with a = 2v—1+s5 = 
2v + (n+3)+ S> [2™-3(n + 2)] and to a super (a,2)-EAT labeling with a =v+1+s = 
m=4 














vt(nt4)+ Y 28m +2)]. 


Theorem 2.2 Forn>1 andr >3,G2T(n+1,n+ 2,2n+4,14,--+ ,n,) admits a super 
(a,1)-EAT labeling with a = 2v+ 8-1 and a super (a,3)-EAT labeling witha =v+s+1, 
where v =|V(G)| and s =3 and nm = 2™-?(n +2) for4d<m<r. 





Proof Let us consider the vertices and edges are defined as in Theorem 2.1. Now, define 
A: V(G) > {1,2,--- ,v} as follows: 


Mc) = 1. For 1 <1; < nj, where i= 1,2,3 and4<i<r: 


30 A.Raheem and A.Q.Baig 


1,41, for u=sz', 
A(u) = (2n+5)—lo, for u= x, 
(4n+9)—le, for u=as, 


and Xai’) = (4n+9) + >> [2-2 (n + 2)] — ly respectively. 
m=4 

The set of all edge-sums {A(x) + A(y) : ey € E(G)} generated by the above formulas forms 

an integer sequence 3,3 + 2,--- ,3+2(e—1), where s = 3. Therefore, by Proposition 1.1, » 

can be extended to a super (a,1)-EAT labeling with a = 2u —1+ s = 2u+ 2 and to a super 

(a, 3)-EAT labeling with a=v+1+s=v+4. 

















As a consequence of Lemma 1.1. and the Theorem 2.1., we have the following corollaries: 





Corollary 2.3 Forn > 1 andr >4, G2 T(n+1,n4+2,2n+4,n4,--- ,n,) admits a super (a, 0)- 
EAT labeling with magic constant a = (3u—n—1)— > [2~3(n4+2)], where Nm = 2™-2(n+2) 


m=4 


for4d<m<r. 


Corollary 2.4 Forn>1andr>4,G2T(n+1,n4+2,2n+4,14,---,n,) admits a super 
(a,2)-EAT labeling with minimum edge weight is a = (20-—n+1)— > [2™-3(n + 2)], where 





m= 


Nm = 2"-2(n + 2) ford<m<r. 


We construct relation between the Super (a, d)-EAT labelings and the (a, d)-EAT labelings 
deduce from Theorem 2.2. and according to the concept of Kotzig and Rosa related to a dual 
labeling, we have the following corollary. 





Corollary 2.5 Forn>1 andr >4,G2T(n+1,n+2, 2n+4,n4,--- ,n,) admits a (a,1)-EAT 
labeling with minimum edge weight is a = 3v and (a,3)-EAT labeling with minimum edge weight 
a= 2u +2, where nm = 2™-2(n + 2) ford<m<r. 





Theorem 2.6 Forn >1 andr >4,G2T7(n+1,n+1,n+4 2,n4,---,n,-) admits a super 
(a,0)-EAT labeling with a = 2v+s—1 and a super (a,2)-EAT labeling witha =v+s+1, 
where v =|V(G)| and 


s=1+ puss + » [2”-4(n + 2) 


m=4 


and nm = 2™-3(n+ 2) for4d<m<r. 


Proof The vertices and edges of the graph G are v = (8n + 4) + S> [2™ -3(n + 2)] and 
m=4 
e =v—1. Define the vertex labeling A: V(G) — {1,2,--- , uv} as follows: 


Super Edge-Antimagic Labeling of Subdivided Star Trees 31 


2 
A(c) = “ ]. For even 1 <1; < nj, where i = 1,2,3 and4<i<r: 





li 
2° 


3 
© 


— pli 
for u= 2, 





Mu)=) 22442 for u= 22, 


poteeee)y _ ls 


a = -tOr u= xs. 


iy _ -3(n+ 2) m—4 Ko ‘ 
A(z;') = ar ea + S- [2™—*(n + 2)] -— 5 respectively. 


m=4 


For odd 1 <1; < nj anda = p32) + >> [2™-4(n + 2)], where i=1,2,3 and 4<i<r: 
m=4 
a+ [243] - 44, for u=a'}, 
Mu) =4 a+ [2¢4] 4 2H, for u= 23, 


a+1+ [Se ) — fet for u= a. 








and 


A(alt) =e +14 pcan) Yh 4 ST p"A4(n+2)] - 4 = 


m=4 





respectively. 


The set of all edge-sums {A(x) + A(y) : cy € E(G)} generated by the above formulas forms 
a consecutive integer sequence (a+1)+1,(a+1)+2,--- ,(a+1)+e, where s = a+2. Therefore, 
by Proposition 2.1, \ can be extended to a super (a,0)-EAT labeling with 


a= 245-1 = 20+ [Et D 4 oti 49) 
m=4 


and to a super (a, 2)-EAT labeling with 


a=v+1l+s=v42+4 [3n4+ 72] + > 2" -4(n 4+ 2)). 


m=4 

















Theorem 2.7 Forn>1 andr >4,G2T7T(n+1,n4+1,n+4 2,n4,---,n,-) admits a super 
(a,1)-EAT labeling with a = 2v+s5—1 and a super (a,3)-EAT labeling witha =v+s+1, 
where v =|V(G)| and s =3 and nm = 2™ 3(n +2) ford<m<r. 


Proof Let us consider the vertices and edges are defined as in Theorem 2.3. Now, we define 
A: V(G) > {1,2,--- ,v} as follows: 


A(c) =n+2. For 1 <1; < nj, where i = 1,2,3 and4<i<r: 


32 A.Raheem and A.Q.Baig 


(n+ 2)—h, for u=sz', 
Au) = 4 (n+2) +h, for u= x, 
3(n+2)—I3, for u=2'g, 

and 


a 


alt) = 8(n + 2) + S- [2”-°(n + 2)] — 1; respectively. 
m=4 

The set of all edge-sums {A(x) + A(y) : cy € E(G)} generated by the above formulas forms 

an integer sequence 3,3 + 2,--- ,3+2(e—1), where s = 3. Therefore, by Proposition 2.1, » 

can be extended to a super (a,1)-EAT labeling with a = 2u —1+ s = 2u+ 2 and to a super 

(a, 3)-EAT labeling with a=v+1+s=v+4. 

















§3. Conclusion 


In this paper, we have proved the super edge anti-magicness of subdivided stars for all possible 
values of d, However the problem of the anti-magicness is still open for different values of magic 


constant. 


References 


1] E.T.Baskoro and A.A.G.Ngurah, On super edge-magic total labelings, Bull. Inst. Combin. 
Appil., 37(2003), 82-87. 

2| E.T.Baskoro, I.W.Sudarsana and Y.M.Cholily, How to construct new super edge-magic 
graphs from some old ones, J. Indones. Math. Soc. (MIHIM), 11:2 (2005), 155-162. 

3] M.Baca, Y.Lin and F.A.Muntaner-Batle, Edge-antimagic labeling of forests, Utilitas Math., 
81(2010), 31-40. 

4| M.Baéa and C. Barrientos, Graceful and edge-antimagic labeling, Ars Combin., 96 (2010), 
505-513. 

5] M.Baéa, P.Kovai, A.Semaniéova -Feniovéfkovaé and M.K.Shafig, On super (a, 1)-edge-antimagic 
total labeling of regular graphs, Discrete Math., 310 (2010), 1408-1412. 

6] M.Baéa, Y.Lin, M.Miller and M.Z. Youssef, Edge-antimagic graphs, Discrete Math., 307(2007), 
1232-1244. 

7| M.Baéa, Y.Lin, M.Miller and R.Simanjuntak, New constructions of magic and antimagic 
graph labelings, Utilitas Math., 60 (2001), 229-239. 

8] M.Baéa, Y.Lin and F.A.Muntaner-Batle, Super edge-antimagic labelings of the path-like 
trees, Utilitas Math., 73 (2007), 117-128. 

9] M.Baéa and M.Miller, Super Edge-Antimagic Graphs, Brown Walker Press, Boca Raton, 
Florida USA, 2008. 

[10] M.Baéa, A.Semanicéova -Fenovéikové and M.K.Shafig, A method to generate large classes 
of edge-antimagic trees, Utilitas Math., 86 (2011), 33-43. 














11 


12 


13 


14 
15 


16 


17 


18 


19 


20 


21 


22 


23 


24 





Super Edge-Antimagic Labeling of Subdivided Star Trees 33 


H.Enomoto, A.S.Llado, T.Nakamigawa and G.Ringle, Super edge-magic graphs, SUT J. 
Math., 34(1980), 105-109. 

R.M.Figueroa-Centeno, R.Ichishima, and F.A.Muntaner-Batle, The place of super edge- 
magic labeling among other classes of labeling, Discrete Math., 231(2001), 153-168. 
R.M.Figueroa-Centeno, R.Ichishima and F.A.Muntaner-Batle, On super edge-magic graph, 
Ars Combin., 64(2002), 81-95. 

J.A.Gallian, A dynamic survey of graph labeling, Elec. J. Combin., 17(2014). 

M.Hussain, E.T.Baskoro and Slamin, On super edge-magic total labeling of banana trees, 
Utilitas Math., 79 (2009), 243-251. 

M.Javaid, M.Hussain, K.Ali and H.Shaker, Super edge-magic total labeling on subdivision 
of trees, Utilitas Math., to appear. 

A.Kotzig and A.Rosa, Magic valuations of finite graphs, Canad. Math. Bull., 13(1970), 
451-461. 

A.Kotzig and A.Rosa, Magic valuation of complete graphs, Centre de Recherches Mathe- 
matiques, Universite de Montreal, 1972, CRM-175. 

S.M.Lee and Q.X.Shah, All trees with at most 17 vertices are super edge-magic, 16th 
MCCCC Conference, Carbondale, University Southern Illinois, November 2002. 
A.A.G.Ngurah, R.Simanjuntak and E.T.Baskoro, On (super) edge-magic total labeling of 
subdivision of Ky,3, SUT J. Math., 43 (2007), 127-136. 

R.Simanjuntak, F.Bertault and M.Miller, Two new (a, d)-antimagic graph labelings, Proc. 
of Eleventh Australasian Workshop on Combinatorial Algorithms, 2000, 179-189. 
K.A.Sugeng, M.Miller Slamin, and M. Baga, (a, d)-edge-antimagic total labelings of cater- 
pillars, Lecture Notes Comput. Sci., 3330(2005), 169-180. 

A.N.M.Salman, A.A.G.Ngurah and N.Izzati, On Super Edge-Magic Total Labeling of a 
Subdivision of a Star S,,, Utilitas Mthematica, 81(2010), 275-284. 

D.B.West, An Introduction to Graph Theory, Prentice-Hall, 1996. 


Math. Combin. Book Ser. Vol.1(2016), 34-41 


Surface Family with a Common Natural Geodesic Lift 


Evren Ergiin 
Ondokuz Mayis University 


Cargamba Chamber of Commerce Vocational School, Cargsamba, Samsun, Turkey 


Ergin Bayram 
Ondokuz Mayis University 


Faculty of Arts and Sciences, Department of Mathematics, Samsun, Turkey 
E-mail: eergun@omu.edu.tr, erginbayram@yahoo.com 


Abstract: In the present paper, we find a surface family possessing the natural lift of a 
given curve as a geodesic. We express necessary and sufficient conditions for the given curve 
such that its natural lift is a geodesic on any member of the surface family. We present 
a sufficient condition for ruled surfaces with the above property. Finally, we illustrate the 


method with some examples. 
Key Words: Ruled surfaces, curve, geodesic, Frenet frame. 


AMS(2010): 53A04, 53A05. 


§1. Introduction 


Curves and surfaces play an important role in differential geometry. In recent years, there is 
an ascending interest on finding surfaces possessing a given curve as a common curve instead 
of finding and characterizing curves on a given surface. In 2004, Wang et. al. [1] proposed 
a method to find surfaces having a given curve as a common geodesic. Kasap et. al. [2] 
generalized the marching-scale functions of Wang and obtained a larger family of surfaces. Li 
et. al. [3] derived the necessary and sufficient constraint for a line of curvature. Bayram et. 
al. [4] studied parametric surfaces which interpolate a given curve as a common asymptotic. 
Ergiin et. al. [5] obtained a surface family from a given spacelike or timelike line of curvature 
in Minkowski 3-space. 

Inspired with the above studies, we find a surface family possessing the natural lift of a 
given curve as a common geodesic. We obtain the sufficient condition for the resulting surface 
to be a ruled surface. 

We start with presenting some background. A parametric curve a(s), [1 < s < La, 
is a curve on a surface P(s,t) in R® that has a constant s or t-parameter value. In this 
paper, a’ denotes the derivative of a with respect to arc length parameter s and we assume 
that @ is a regular curve with a” (s) 4 0, Li < s < Ly. For every point of a(s), the set 


lReceived May 12, 2015, Accepted February 8, 2016. 


Surface Family with a Common Natural Geodesic Lift 35 


{T (s),N(s),B(s)} is called the Frenet frame along a(s), where T(s) = a’(s), N(s) = 
Tal and B(s) =T(s) x N(s) are the unit tangent, principal normal, and binormal vectors of 


the curve at the point a(s), respectively. Derivative formulas of the Frenet frame is governed 
by the relations 


j T (s) 0 k (s) 0 T (s) 
ae N(s) | = | —«(s) 0 T (s) N(s) |, (1) 
B(s) 0 —t(s) 0 B(s) 
where « (s) = |ja” (s)|| and 7 (s) = — (B’(s), N(s)) are called the curvature and torsion of the 


curve a(s), respectively [6]. 


Let M be a surface in R® and let a : J —+ M be a parameterized curve. a is called an 
integral curve of X if 


X (a(s)) (for allt € J), 


S| 
Ww 
— 
Q 
— 
wD 
wa 
— 
I 


where X is a smooth tangent vector field on M. We have 


TM = |) TpM= x(M), 
PEM 


where TpM is the tangent space of M at P and y (MM) is the space of tangent vector fields on 
M. 


For any parameterized curve a: 1 —> M ,a@:I —+TM given by ([7]) 


, 


& (s) = (a(s),a" (8)) =a" (5) lays) (2) 


is called the natural lift of a on TM. 


If a rigid body moves along a unit speed curve a (s), then the motion of the body consists 
of translation along a and rotation about a. The rotation is determined by an angular velocity 
vector w which satisfies T’ = w x T, N’=w x N and B’ =w x B. The vector w is called the 
Darbouz vector. In terms of Frenet vectors T, N and B, Darboux vector is given by w = TT +KB 
[8]. Also, we have «& = ||w||cos6, 7+ = ||w||sin@, where @ is the angle between the Darboux 
vector w and binormal vector B(s) of a. Observe that @ = arctant (Fig. 1). 





Fig.1 Darboux vector w, tangent vector TJ’ and binormal vector B of a 


Let a(s), Ly < s < Lg, be an arc length curve and @(s), Lyi < s < Lg, be the natural 


36 Evren Ergtin and Ergin Bayram 


lift of a. Then we have 


T (s) 0 t* 0 T (s) 
N(s) | =| —cos@ 0. sin@ N(s) |, (3) 
B(s) sinO 0 cosé B(s) 


where {T'(s),N (s),B(s)} and {T (s),N(s), B(s)} are the Frenet frames of the curves a and 
a, respectively, and @ is the angle between the Darboux vector and binormal vector of a. 


§2. Surface Family with a Common Natural Geodesic Lift 


Suppose we are given a 3-dimensional parametric curve a(s), Ly < s < Lg, in which s is the 
arc length and |la”’ (s)|| 40, Ly <s < Lg. Let a@(s), Li < 5 < Le, be the involute of a(s). 
Surface family that interpolates @(s) as a common curve is given in the parametric form 


as 


P(s,t) = @(s) + u(s,t)T (s) + v(s,t) N(s) +w(s,t) B(s), (4) 


Iy <s<L2, Ty <t < To, where u(s,t), v(s,t) and w(s,t) are Ct functions and are called 
marching-scale functions and {T (s),N (s),B(s)} is the Frenet frame of the curve a. Using 
Eqn. (3) we can express Eqn. (4) in terms of Frenet frame {T (s), N (s),B(s)} of the curve a 
as 


P(s,t) = @(s)+(w(s,t) sin@ — v(s,t)cos@) T (s) (5) 
+u (s,t) N (s) + (v(s,t)sin@ + w (s,t) cos@) B(s), 


where Ly <s< lng, Ti <t<T. 


Remark 1 Observe that choosing different marching-scale functions yields different surfaces 
possessing @(s) as a common curve. 


Our goal is to find the necessary and sufficient conditions for which the curve @(s) is 
isoparametric and geodesic on the surface P (s,t). Firstly, as @(s) is an isoparametric curve on 
the surface P (s,t), there exists a parameter to € [T1, T>] such that 


u(s, to) = vu (s, to) = w (s, to) = 0, Ty <s< Le, T; < to SS To. (6) 


Secondly the curve @ is geodesic on the surface P(s,t) if and only if along the curve the 
surface normal vector field n (s, to) is parallel to the principal normal vector field N of the curve 


&. The normal vector of P (s,t) can be written as 


OP (s,t) ig OP (s,t) 


Re Dt 


Surface Family with a Common Natural Geodesic Lift 37 


By Eqns. (3) and (5), the normal vector along the curve @ can be expressed as 


1(5;t0) = 8 | SE (s,t0) N (s) +2 (sto) B(s)) (0 


where « is the curvature of the curve a. Since k(s) #0, Li <5 < In, the curve @ is a geodesic 
on the surface P (s,t) if and only if 


a) 0 
i (s, to) £0, HT (s,to) =0. 


So, we can present: 


Theorem 2 Let a(s), [1 <8 < Lo, be a unit speed curve with nonvanishing curvature and 
a(s), Ly <8 < Lo, be its natural lift. @(s) is a geodesic on the surface (4) if and only if 


u(s,to) =v(s,to) = w(s,to) =0, 


8 
BW (5, to) £0, & (s,t9) =0 . 
ot » 40 > Oe \er%O ’ 


where Ly <s< Lo, Ti <t, to < To (to fixed). 


Corollary 3 Let a(s), [1 <5 < Le, be a unit speed curve with nonvanishing curvature and 
a(s), Ly <8 < Le, be its natural lift. If 


u(s,t) =w(s,t) = (¢—to), v(s,t) =0, (9) 


where Ly <s < Le, Ty <t,to < To (to fixed) then (4) is a ruled surface and & is a geodesic on 
at. 


§3. Examples 


Example 1 Let a(s) = ($coss,1—sins,—2coss) be a unit speed curve. Then, it is easy to 
show that 


4 
T(s) = = (-$sins,- coss, Sins), 


-( 
+ Ge 


9 


4 3 
5 C088, sin s, » 5 008s J , 


By 
50 


T=0, 0. 








We have 
aa 4, 3, 
a(s)= 5 sins, oss, - sins 


38 Evren Ergtin and Ergin Bayram 


as the natural lift of a@ with Frenet vectors 


_ 4 3 

T(s) = (- coss, sin, 3 c088 ‘ 
— 4 

N(s) = (3 sins, cos, ~2 sin ; 


oG)= (-3.0,-3). 


If we choose u(s,t) = w(s,t) =t, u(s,t) = 0, then Eqn. (9) is satisfied and we get the 
ruled surface 


P,(s,t) = a@(s)+t[T(s) + B(s)| 


“ 
4, 3 jo 
= —¢ (sins + teoss) — Ft, tsins — coss, 
ag +t ) = 
=(sins+tcoss)— <t), 
5 5 


—2<s<2, -—1<t< 1, possessing @ as a geodesic such as those shown in Fig.2. 





Fig.2 Ruled surface P;(s;t) as a member of the surface family and 


its common natural geodesic lift & 


For the same curve, if we choose u(s,t) =e? —1, v(s,t) =0, w(s,t) =t, then Eqn. (8) 
is satisfied and we obtain the surface 
Py, (s, t) = 


(s) + (e%* —1) T'(s) +tB(s) 
( 


-: (c* — 1) coss + sins) — st (c** — 1) sins —coss, 


opwiAam~N ei 


((e** — 1) coss + sins) — =) . 


Surface Family with a Common Natural Geodesic Lift 39 


where —3 <5 <3, —1<t< 1 interpolating @ as the natural geodesic lift (Fig. 3). 





Fig.3 P2(s;t) as a member of the surface family and 


its common natural geodesic lift & 


Example 2 Let a(s) = (4 sin s, 5, v8 cos s) be an arc length helix. One can show that 


T(s) = aD ee. TS ay 
8s) = = fay s], 


N(s) = (-sins,0,—coss), 

B(s) = (-- 3 
8) = see 5 gems ; 

J/3 1 7 

oe oe 


We obtain 


2 "2? 2 


7 (2 i a3. 
a&(s) = | —coss, ~,-——sins 


T(s) = (—sins,0,—coss), 
N(s) = (-—coss,0,sins), 
B(s) = (0,1,0). 


Choosing marching scale functions as u(s,t) = s?t, v(s,t) = 0, w(s,t) = sint we get the 


40 


surface 


P3 (s, t) 


Evren Ergiin and Ergin Bayram 


a(s) +s*tT (s) + sintB(s) 


3 1 3 
(2 cos s — s*tsins, 5 + sint, 2 sin s — Hteos) 


satisfying Eqn. (8) possessing @ as a common natural geodesic lift (Fug. 4). 







= 


BESS. 
TY 
uth 











Fig.4 P3(s;t) as a member of the surface family and 


its common natural geodesic lift & 


If we let u(s,t) = stant, v(s,t) = (cost) —1, w(s,t) = ssint, then Eqn. (8) is satisfied 


and we have 


Py (s, t) 


a@(s) + stantT (s) + (cost — 1) N(s) + ssintB (s) 


3 1 
(2 cos s — s (tant) sins + coss (1 — cost), 5 + ssint, 


3 
me sin s — s (tant) cos s + sin s (cost — ») ; 


0<s <3, 0<t< 1, as a member of the surface family possessing @ as a common natural 


geodesic lift shown in Fig.5. 


Surface Family with a Common Natural Geodesic Lift 41 






=—SANY 
SQXx 












ody 
AGH 
Zeeei < 
SE JE 
A Ze 
ZZ 
Zu) 
Zu, 


Fig.5 P,(s,t) as a member of the surface family and 


its common natural geodesic lift @. 


Acknowledgments 


The second author would like to thank TUBITAK (The Scientific and Technological Research 
Council of Turkey) for their financial supports during his doctorate studies. 


References 

1] G.J.Wang, K.Tang and C.L.Tai, Parametric representation of a surface pencil with a com- 
mon spatial geodesic, Comput. Aided Des., 36 (5) (2004), 447-459. 

2] E.Kasap, F.T. Akyildiz and K.Orbay, A generalization of surfaces family with common 
spatial geodesic. Appl. Math. Comput 

3] C.Y. Li, R.H. Wang and C.G.Zhu, Parametric representation of a surface pencil with a 
common line of curvature. Comput. Aided Des., 43 (9) (2011), 1110-1117. 

4) E.Bayram, F.Gtler and E.Kasap, Parametric representation of a surface pencil with a 
common asymptotic curve. Comput. Aided Des., 44 (2012), 637-643. 

5] E.Ergtin, E.Bayram and E.Kasap, Surface pencil with a common line of curvature in 





oO 





Minkowski 3-space. Acta Math. Sinica, English Series. 30 (12) (2014), 2103-2118. 

M.P. do Carmo, Differential geometry of curves and surfaces, Englewood Cliffs, New Jersey 
1976. 

J.A. Thorpe, Elementary topics in differential geometry, Springer-Verlag, New York 1979. 
J. Oprea, Differential geometry and its applications , Pearson Education Inc., USA 2006. 


Math.Combin.Book Ser. Vol.1(2016), 42-56 


Some Curvature Properties of LP-Sasakian Manifold with 


Respect to Quarter-Symmetric Metric Connection 


Santu Dey and Arindam Bhattacharyya 


(Department of Mathematics, Jadavpur University, Kolkata-700032, India) 


E-mail: santu.mathju@gmail.com, bhattachar1968@yahoo.co.in 


Abstract: The objective of the present paper is to study the curvature tensor of the 
quarter-symmetric metric connection with respect to Lorentzian Para-Sasakian manifold 
(briefly, Z.P-Sasakian manifold). It is shown that if in the manifold M”, W. = 0, then 
the manifold M” is locally isomorphic to $”(1), where We is the W2-curvature tensor of 
the quarter-symmetric metric connection in a LP-Sasakian manifold. Next we study gen- 
eralized projective ¢-Recurrent LP-Sasakian manifold with respect to quarter-symmetric 
metric connection. After that ¢-pseudo symmetric LP-Sasakian manifold with respect to 
quarter-symmetric metric connection is studied and we also discuss LP-Sasakian manifold 
with respect to quarter-symmetric metric connection when it satisfies the condition P.S =0, 
where P denotes the projective curvature tensor with respect to quarter-symmetric metric 
connection. Further, we also study €-conharmonically flat LP-Sasakian manifold with re- 
spect to quarter-symmetric metric connection. Finally, we give an example of [P-Sasakian 


manifold with respect to quarter-symmetric metric connection. 


Key Words: Quarter-symmetric metric connection, W2-curvature tensor, generalized pro- 
jective ¢-recurrent manifold, ¢-pseudo symmetric LP-Sasakian manifold, projective curva- 


ture tensor, €-conharmonically flat LP-Sasakian manifold. 


AMS(2010): 53025, 53C15. 


§1. Introduction 


The idea of semi-symmetric linear connection on a differentiable manifold was introduced by 
Friedmann and Schouten ([1]). Further, Hayden ([3]), introduced the idea of metric connection 
with torsion on a Riemannian manifold. In ({16]), Yano studied some curvature conditions for 
semi-symmetric connections in Riemannian manifolds. 

The quarter-symmetric connection generalizes the semi-symmetric connection. The semi- 
symmetric metric connection is important in the geometry of Riemannian manifolds having 
also physical application; for instance, the displacement on the earth surface following a fixed 


1The first author is supported by DST ‘INSPIRE’ of India. 
2Received July 16, 2015, Accepted February 12, 2016. 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 43 


point is metric and semi-symmetric. 


In 1975, Golab ([2]) defined and studied quarter-symmetric connection in a differentiable 
manifold. 


A linear connection V on an n-dimensional Riemannian manifold (M” is said to be a 
@g 


quarter-symmetric connection [2] if its torsion tensor T' defined by 
T(X,Y) =VxY —VyX — [X,Y], (1:1) 


is of the form 


T(X,Y) = (VY )oX — (X) oY, (1.2) 


where 7 is a non-zero 1-form and ¢ is a tensor field of type (1,1). In addition, if a quarter- 


symmetric linear connection V satisfies the condition 
(Vxg)(¥,Z) =0 (1.3) 


for all X,Y, Z € x(M), where x(M) is the set: of all differentiable vector fields on M, then V is 
said to be a quarter-symmetric metric connection. In particular, if 6X = X and ¢Y = Y for all 
X,Y € x(M), then the quarter-symmetric connection reduces to a semi-symmetric connection 


[1]. 


On the other hand Matsumoto ([5]) introduced the notion of LP-Sasakian manifold. Then 
Mihai and Rosoca([9]) introduced the same notion independently and obtained several results 
on this manifold. UP-Sasakian manifolds are also studied by Mihai([9]), Singh({15]) and others. 


Definition 1.1 A LP-Sasakian manifold is said to be generalized projective @-recurrent if its 


curvature tensor R satisfies the condition 
¢°((VwP)(X,Y)Z) = A(W)P(X,Y)Z + B(W)[g(¥, Z)X — oY, Z)X], (1.4) 
where A and B are 1-forms, 3 is non-zero and these are defined by 
A(W) = g(W, pi), B(W) = g(W, pa), 


and where p; and p2 are vector fields associated with 1-forms A and B respectively and P is 


the projective curvature tensor for an n-dimensional Riemannian manifold M, given by 
1 
P(X,Y)Z = R(X, Y)Z—- are olee Z)X — S(X,Z)Y], (1.5) 
n— 


where R and §S are the curvature tensor and Ricci tensor of the manifold. 


Definition 1.2 A LP-Sasakian manifold (M”, ¢,&,7,g)(n > 2) is said to be 6-pseudosymmetric 


44 Santu Dey and Arindam Bhattacharyya 


([4]) of the curvature tensor R. satisfies 


¢°((VwR)(X,Y)Z) 


I 


2A(W)R(X,Y)Z + A(X)R(W,Y)Z 
A(Y)R(X,W)Z + A(Z)R(X,Y)W 
GR(X,Y)Z,W)p (1.6) 


+ + 


for any vector field X,Y, Z and W, where p is the vector field associated to the 1-form A such 
that A(X) = g(X,p). In particular, if A = 0 then the manifold is said to be ¢-symmetric. 


After Golab(([2]), Rastogi ([13], [14]) continued the systematic study of quarter-symmetric 
metric connection. In 1980, Mishra and Pandey ([8]) studied quarter-symmetric metric con- 
nection in a Riemannian, Kaehlerian and Sasakian manifold. In 1982, Yano and Imai((17]) 
studied quarter-symmetric metric connection in Hermition and Kaehlerian manifolds. In 1991, 
Mukhopadhyay et al.({10]) studied quarter-symmetric metric connection on a Riemannian man- 
ifold with an almost complex structure ¢. However these manifolds have been studied by many 
geometers like K. Matsumoto ([6]), K. Matsumoto and I. Mihai ([8]), I. Mihai and R. Rosca([5]) 
and they obtained many results on this manifold. 

In 1970, Pokhariyal and Mishra ([11]) have introduced new tensor fields, called W2 and 
&-tensor fields in a Riemannian manifold and studied their properties. Again, Pokhariyal 
({12]) have studied some properties of these tensor fields in a Sasakian manifolds. Recently, 
Matsumoto, Ianus and Mihai ([6]) have studied P-Sasakian manifolds admitting W2 and E- 
tensor fields. The W2-curvature tensor is defined by 


W2(X,Y)Z = R(X, Y)Z+ —*_{9(X, Z)QY — G(Y, Z)QX}, (1.7) 


where R and Q are the curvature tensor and Ricci operator and for all X,Y, Z € x(M). 
The conharmonic curvature tensor of LP-Sasakian Manifold M” is given by 


C(X,Y)Z = R(X,Y)Z- —*i0¥, Z)QX — g(X,Z)QY 
fh. SVE Z es SIZ: (1.8) 


where R and S are the curvature tensor and Ricci tensor of the manifold. 

Motivated by the above studies, in the present paper, we consider the W2-curvature ten- 
sor of a quarter-symmetric metric connection and study some curvature conditions. Section 
2 is devoted to preliminaries. In third section, we find expression for the curvature tensor, 
Ricci tensor and scalar curvature of DL P-Sasakian manifold with respect to quarter-symmetric 
metric connection and investigate relations between curvature tensor (resp. Ricci tensor) with 
respect to the semi-symmetric metric connection and curvature tensor (resp. Ricci tensor) 
with respect to Levi-Civita connection. In section four, W2 curvature tensor with respect to 
quarter-symmetric metric connection is studied. In this section, it is seen that if W2 = 0in 
M”, then M” is locally isomorphic to $”(1), where W2 is curvature tensor with respect to 
quarter-symmetric metric connection VY. Next we have obtained some expression of Ricci ten- 
sor when (W2(€, Z).$)(X,Y) = 0 in LP-Sasakian manifold with respect to quarter-symmetric 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 45 


metric connection. In section five deals with generalized projective ¢-Recurrent DL P-Sasakian 
manifold with respect to quarter-symmetric metric connection. In section six, ¢-pseudo sym- 
metric LP-Sasakian manifold with respect to quarter-symmetric metric connection is studied. 
In next section, we cultivate [P-Sasakian manifold with respect to quarter-symmetric metric 
connection satisfying when it satisfies the condition P.S = 0, where P denotes the projec- 
tive curvature tensor with respect to quarter-symmetric metric connection. Finally, We study 
€-conharmonically flat DP-Sasakian manifold with respect to quarter-symmetric metric con- 


nection. 


§2. Preliminaries 


A n-dimensional, (n = 2m + 1), differentiable manifold M” is called Lorentzian para-Sasakian 
(briefly, L.P-Sasakian) manifold ((5], [7]) if it admits a (1,1)-tensor field ¢, a contravariant 
vector field €, a 1-form 7 and a Lorentzian metric g which satisfy 


n(g) = —1, (2:1) 

PX =X+U(XYE, (2.2) 

HX, OY) = Gg X,Y) +0(X)n(Y), (2.3) 

IX, §) = 0(X), (2.4) 

Vxt =X, (2.5) 

(VxO)(Y) = G(X, YE + n(V)X + 2n(X)n(V)E, (2.6) 


where, V denotes the covariant differentiation with respect to Lorentzian metric g. It can be 
easily seen that in an L.P-Sasakian manifold the following relations hold: 


~& = 0, (PX) =0, (2.7) 
rank(¢) =n—1. (2.8) 

If we put 
O(X,Y) =g(X,4Y), (2.9) 


for any vector field X and Y, then the tensor field ®(X,Y) is a symmetric (0, 2)-tensor field 
([5]). Also since the 1-form 77 is closed in an LP-Sasakian manifold, we have (([5]) 


(Vxn)(Y) = (X,Y), O(X, 6) =0 (2.10) 


for all X,Y € y(M). 
Also in an LP-Sasakian manifold, the following relations hold ([7]): 


WR(X,Y)Z,§) = (R(X, Y)Z) = g(¥, Z)(X) — G(X, Z)n(¥), (2.11) 


46 Santu Dey and Arindam Bhattacharyya 


R(X,Y)E = n(V)X — (XY, 


( 

( 
RE, X)E=X + (XE, (2. 

( 

( 


bo 
_ 
aw 
eee 


S(X, €) = (n— 1)n(X), 
QX = (n—-1)X,r=n(n-1), 


where Q is the Ricci operator, i.e. 
g(QX,Y) = S(X,Y) (2.17) 
and r is the scalar curvature of the connection V. Also 
S(oX, PY) = S(X,Y) + (n— 1)n(X)n(Y), (2.18) 


for any vector field X, Y and Z, where R and S are the Riemannian curvature tensor and Ricci 
tensor of the manifold respectively. 


§3. Curvature tensor of L P-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 


In this section we express R(X ,Y)Z the curvature tensor with respect to quarter-symmetric 
metric connection in terms of R(X,Y)Z the curvature tensor with respect to Riemannian 


connection. 


Let V be the linear connection and V be Riemannian connection of an almost contact 
metric manifold such that 
VxY =VxY+L(X,Y), (3.1) 


where L is the tensor field of type (1,1). For V to bea quarter-symmetric metric connection 
in M™, we have ([2]) 


HK 


L(X,Y) = 5T(X, YET yY er wx, (3.2) 


and 
g(T'(X,Y), Z) = 9(T(X,Y), Z). (3.3) 


From the equation (1.2) and (3.3), we get 
T'(X,Y) = n(X)oY + g(OX, VE. (3.4) 
Now putting the equations (1.2) and (3.4) in (3.2), we obtain 


L(X,Y) = (¥)bX + g(@X, VE. (3.5) 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 47 


So, a quarter-symmetric metric connection V in an LP-Sasakian manifold is given by 
VxY =VyX + n(V)OX + G(bX, VIE. (3.6) 


Thus the above equation gives us the relation between quarter-symmetric metric connection 


and the Levi-Civita connection. 


The curvature tensor R of M” with respect to quarter-symmetric metric connection V is 
defined by 
R(X, Y)Z =VxVyZ -—VyVxZ — VixyyZ. (3.7) 


A relation between the curvature tensor of M with respect to the quarter-symmetric metric 
connection V and the Riemannian connection V is given by 


R(X,Y)Z = R(X,Y)Z+9(oX,Z)bY — g(GY, Z)OX 
+ MZ )inV)x — XY} + {9¥, Z)(X) — g(X, Zn VE, (3.8) 


where R and R are the Riemannian curvature tensor with respect to V and V respectively. 


From the equation (3.8), we get 


S(Y,Z) = S(Y,Z) + (n—1)n(Y)n(Z), (3.9) 


where $ and § are the Ricci tensor with respect to V and V respectively. This gives 


QY = QY + (n- 1)n (VDE. (3.10) 
Contracting (3.9), we obtain, 
f=r—(n-1)), (3.11) 
where # and r are the scalar curvature tensor with respect to V and V respectively. Also we 
have 
R(X,Y)E =0, (3.12) 
which gives 
m(R(X,Y)E) = 0, (3.13) 
and 
R(E,Y)Z = 0, (3.14) 
which gives 
mMR(E,Y)Z) = 0. (3.15) 


§4. W2-Curvature Tensor of LP-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 


The W.-curvature tensor of [ P-Sasakian manifold MM” with respect to quarter-symmetric met- 


48 Santu Dey and Arindam Bhattacharyya 


ric connection V is given by 
= e 1 ~ 2 
Wa(X,Y)Z = R(X,Y)Z + —{ 9X, ZY — GV, Z)OX}. (4.1) 
Using the equations (3.8) and (3.10) in (4.1), we get 


Wo(X,Y)Z= = R(X,Y)Z + 9(bX, Z)dY — g(bY, Z)OX 

+ (Z){n(V)X — {XV} 

+ {9 (¥, Z)n(X) — 9(X, Z)n(V FE 

+ so X, Z{Q¥ + (n—V)n(V)G 

= HY, ZMOX + (nn X )E}]- (4.2) 





Now using the equation (1.7) in (4.2), we obtain 


Wo(X,Y)Z= = W2(X,Y)Z + 9(bX, Z)Y — 9(bY, Z)OX 
+ (Z){n(¥)X — (X)V} 

+ {g9(¥, Z)n(X) — G(X, Z)n(¥ FE 

+S [9(X,Z)(n—1)nl¥)E 

— 9 ¥, Z)(n — 1)n(X)€]. (4.3) 





Putting Z = € in (4.3) and using the equations (2.1), (2.4), (2.7) and (1.7), we get 
which gives 
m(W2(X,¥)g) = 0. (4.5) 


Again putting X = € in (4.3) and using the equations (2.1), (2.4), (2.7), (2.12) and (1.7), 


we get 
Wal,Y)Z = n(Z)¥ + n(¥)n(Z)E. (4.6) 


This gives 
m(W2(f,¥)Z) = 0. (4.7) 
Theorem 4.1 In LP-Sasakian Manifold M”, if the W2-Curvature tensor of with respect to 
quarter-symmetric metric connection vanishes, then it is locally isomorphic to S"(1). 
Proof Let Wz = 0. From the equation (4.2), we have 
R(X,Y)Z = g(PY, Z)OX — g(PX, Z)OY + (Z){n(X)¥ — n(¥)X} 
1 
+ {9(X, Z)nY) — GV, Zn XE — — FIX ZHLQY + (n — 1) nV) 8} 
— GV, Z){QX + (nr — 1)n(X) 4). (4.8) 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 49 


Taking the inner product of the above equation and using (2.1), (2.4), (2.7), we get 
MR(X,Y)Z) = {g(¥, Z)X — g(X, ZY}, (4.9) 


which gives 
R(X, Y,Z,U) = {g(Y, Z)g(X,U) — g(X, Z)g(Y,U)}. (4.10) 


This shows that M” is a space of constant curvature is 1, that is, it is locally isomorphic 
to S”(1). 














Suppose let (Wo(€, Z).S)(X,Y) = 0. This gives 
Now using the equation (3.9) in (4.11), we get 


S(Wa(E, Z)X,Y) + (n— 1)n(WalE, Z)X)n(¥) 
S(X, Wa(é, Z)Y) + (n— 1)n(Wa(E, Z)Y)n(X) = 0. (4.12) 


Using the equation (2.15), (4.6) and (4.7) in (4.12), we obtain 


MX)S(Y,Z) + (n—I)n(X)n(¥)n(Z) + 0(V)S(X, Z) 
+ (n= 1)n(X)n(¥)n(Z) = 0. (4.13) 





Putting X = € and using the equation (2.1) and (2.4) in (4.13), we get 
S(Y, Z) = (1 —n)n(¥)n(Z). (4.14) 


So, we have the following theorem. 


Theorem 4.2 A LP-Sasakian manifold M” with respect to quarter-symmetric metric connec- 
tion V_ satisfying (Wo(€, Z).S)(X,Y) =0 is the product of two 1-forms. 


§5. Generalized Projective ¢-Recurrent [ P-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 


The projective curvature tensor for an n-dimensional Riemannian manifold M with respect to 


quarter-symmetric metric connection is given by 


1 


7/8, 2)X - §(%, ZY], (5.1) 


P(X,Y)Z = R(X,Y)Z—- 


where R and S are the curvature tensor and Ricci tensor of the manifold. 


Let us consider generalized projective ¢-recurrent LP-Sasakian manifold with respect to 


50 Santu Dey and Arindam Bhattacharyya 


quarter-symmetric metric connection. By virtue of (1.4) and (2.2), we get 


(VwP\(X,Y)Z + ((VwP)(X,Y)Z)E = AWW) P(X, Y)Z 
+ BW)[g(¥, Z)X — g(X, Z)Y], (5.2) 





from which it follows that 





g((VwP)(X,Y)Z,U) + ((VwP)(X,Y)Z)nU) = A(W)9(P(X,Y)Z,U) 
+ B(W)[g(¥, Z)9(X,U) — 9({X, Z)g(¥,U)]. (5.3) 
Let {e;}, i = 1,2,--- ,n be an orthonormal basis of the tangent space at any point of the 


manifold. Then putting X = U = e; in (5.3) and taking summation over i, 1 < i < n, we get 





(wSyx,u) — WE gx.) + PW) — Gy 5x, On(U) 
+ SBE x n(y) - POE a) 
= AW) 8(x,U) - 9(X,0) 
4+ InB(W)g(X,U). (5.4) 


Putting U = € in (5.4) and using the equation (3.6), (3.9) an (3.11), we obtain 


ik 





A(W)[L - nl X) + (n — BOW) n(X) = 0. (5.5) 
Putting X = € in (5.5), we get 
BOW) = AW). (5.6) 


Thus we can state the following theorem. 


Theorem 5.1 In a generalized projective d-ecurrent LP-Sasakian manifold M" (n > 2), the 
1-forms A and B are related as (5.6). 


§6. ¢-Pseudo Symmetric LP-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 
Definition 6.1 A ZP-Sasakian manifold (M”, ¢, €,7,g)(n > 2) is said to be ¢-pseudosymmetric 


with respect to quarter symmetric metric connection if the curvature tensor R satisfies 


’((VwR)(X,Y)Z) = 2A(W)R(X,Y)Z + A(X)R(W,Y)Z 
+ A(Y)R(X,W)Z + A(Z)R(X,Y)W + g(R(X,Y)Z,W)p (6.1) 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 51 


for any vector field X, Y, Z and W, where p is the vector field associated to the 1-form A such 
that A(X) = g(X, p). Now using (2.2) in (6.1), we have 


(VwR)(X,Y)Z + n((VwR)(X,Y)Z)E = 2A(W) R(X, Y)Z 
(X)R(W,Y)Z + A(Y) R(X, W)Z 
(Z)R(X,Y)W + g(R(X,Y)Z,W)p. (6.2) 


as? JA 
A 





From which it follows that 


g(VwRY(X,Y)Z,U) + ((VwR)(X,Y)Z)n(U) = 2A(W)g(R(X, Y)Z,U) 
+ A(X)g(R(W,Y)Z,U) + A(Y)g(R(X, W)Z,U) 
+ A(Z)g(R(X,Y)W,U) + g(R(X,Y)Z,W)A(U). (6.3) 








Let {e; : i = 1,2,--- ,n} be an orthonormal basis of the tangent space at any point of the 
manifold. Setting X = U = e; in (6.3) and taking summation over i, 1 <7 <n, and then using 
(2.1), (2.4) and (2.7) in (6.3), we obtain 


(VwS)(¥,Z) + 9((VwR)(E,Y)Z,€) =2A(W)S(Y, Z) 
+ A(Y)S(W,Z) + A(Z)S(Y,W) 
+ A(R(W,Y)Z) + A(R(W, Z)Y). (6.4) 





By virtue of (3.14) it follows from (6.4) that 


(VwS)(Y,Z) = 2A(W)S(Y,Z) + A(Y)S(W, Z) + A(Z)S(Y,W) 
+ A(R(W,Y)Z) + A(R(W, Z)Y). (6.5) 


So, we have the following theorem: 


Theorem 6.1 A ¢-pseudo symmetric LP-Sasakian manifold with respect to quarter-symmetric 
metric connection is pseudo Ricci symmetric with respect to quarter sym- metric non-metric 


connection if and only if 


A(R(W,Y)Z) + A(R(W, Z)Y) = 0. 


§7. LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric 
Connection Satisfying P.S = 0. 
A LP-Sasakian manifold with respect to the quarter-symmetric metric connection satisfying 
(P(X, Y).5)(Z,U) =0, (7.1) 


where S is the Ricci tensor with respect to a quarter-symmetric metric connection. Then, we 


52 Santu Dey and Arindam Bhattacharyya 


have 


S(P(X,Y)Z,U) + §(Z, P(X, Y)U) = 0. (7.2) 


Putting X = € in the equation (7.2), we have 
S(P(E,Y)Z,U) + $(Z, P(E, YU) =0. (7.3) 
In view of the equation (5.1), we have 
PEY)Z = REY)Z - 18, 26 - 82] (7.4) 
for X,Y, Z € x(M). 
Using equations (3.9) and (3.14) in the equation (7.4), we get 
PEY)Z = -—1S(Y, 2) + (n — Inn ZS. (7.5) 


Now using the equation (7.5) and putting U = € in the equation (7.3) and using the 
equations (2.2), (2.15) and (3.9) we get 


S(Y,Z) + (n— 1)n(Y)n(Z) = 0. (7.6) 


S(Y, Z) = —(n— 1)n(¥ )n(Z). (7.7) 


In view of above discussions we can state the following theorem: 


Theorem 7.1 A n-dimensional LP-Sasakian manifold with a quarter-symmetric metric con- 


nection satisfying P.S =0 is the product of two 1-forms. 


§8. €-Conharmonically Flat [P-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 


The conharmonic curvature tensor of LP-Sasakian manifold M” with respect to quarter- 


symmetric metric connection V is given by 


COGVIZ” = REGY)Z= —* 10, DOK = 9G 2)OY 


de SOB OR AY (8.1) 


where R and S$ are the curvature tensor and Ricci tensor with respect to quarter-symmetric 


metric connection. 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 53 


Using (3.8), (3.9) and (3.10) in (8.1), we get 


C(X,Y)Z 


+ MZ){n(V)X — 1(X)¥} 


+ {9(¥,Z)n(X) — 9X, ZV} 
— (9X, Z){Q¥ + (n—n(V)4} 


- al’. Z){QX + (n— 1)(X)E}] 
_ —slo¥, Z){QX + (n= 1)n(X)G 


— G(X, Z){QY + (n — 1)n(¥)e} + SY, 2) X 
+ (n—1)n(Y)n(Z)X — S(X,Z)Y 
— (n= 1)n(X)n(Z)Y}. (8.2) 





R(X, Y)Z + g(oX, Z)OY — g(GY, Z)oX 
( 


C(X,Y)Z = C(X,Y)Z + g(oX, Z)bY — g( bY, Z)bX 
Z) MX — n(X)V} + {9V, Z)n(X) 
n(V)}e — "Shay, Z)n(xXYE 
nV )E + (¥ )n(Z)X 
)Y], (8.3) 


where C is given in (1.8). Putting Z = € in (8.3) and using (2.1), (2.4) and (2.7), we obtain 





CULY)E = C(X,Y)E- (00) X - OY} 
— 2Sinoy - nv). (8.4) 


Suppose X and Y are orthogonal to €, then from (8.4), we obtain 
C(X, YE = C(X, VE. (8.5) 
So, by the above discussion we can state the following theorem: 


Theorem 8.1 An n-dimensional LP-Sasakian manifold is €-conharmonically flat with respect 
to the quarter-symmetric metric connection if and only if the manifold is also €-conharmonically 
flat with respect to the Levi-Civita connection provided the vector fields X and Y are orthogonal 
to the associated vector field €. 


§9. Example 3-Dimensional LP-Sasakian Manifold with Respect to 


Quarter-Symmetric Metric Connection 


We consider a 3-dimensional manifold M = {(a,y,u) € R*}, where (a, y,u) are the standard 


54 Santu Dey and Arindam Bhattacharyya 


coordinates of R?. Let e1,€2,e€3 be the vector fields on M? given by 


3) ) 


U—ZX 
e3 = —-—. 
Ou 


e, = —e” eg = —e" 7, 
Oy 


0 
Ox’ 

Clearly, {e1, €2,e3} is a set of linearly independent vectors for each point of M and hence 
a basis of y(M). The Lorentzian metric g is defined by 


g(e1, 2) = g(e2,e3) = g(e1, €3) = 0, 
=1 


g(é1, €1) , g(e2,e2) =1, g(es,e3) = —1. 


Let 7 be the 1-form defined by 7(Z) = g(Z,e3) for any Z € y(M) and the (1,1) tensor 
field @ is defined by 
ger =—€1, gez = —e2, Ge3 = 0. 


From the linearity of ¢ and g, we have 


n(e3) =-1, 
PX =X +n(Xeg 


and 
GPX, PY) = G(X, Y) +7(X)n(Y) 


for any X € y(M). Then for e3 = €, the structure (¢,,7,g) defines a Lorentzian paracontact 
structure on M. Let V be the Levi-Civita connection with respect to the Lorentzian metric 
g. Then we have 


[e1, €2] = —e"ea, [e1,e3] = —e1, [e2, e3] = —e2. 
Koszul’s formula is defined by 


2g(VxY,Z) = XgG(Y,Z)+Yq(Z, xX) -— Zg(X,Y) 
Os [y Z)) Fs GY, xs Z)) + g(Z, [X, Y]). 


Then from above formula we can calculate followings: 


Ve €1 = €3, Ve, €2 = 0, Ve €3 = —e€2, 
Ver€1 = —e"€2, Vere2 = —€3 — €"€1, Vere3 = —€2; 
Ve3€1 = 0, Ve3€2 => 0, V eg €3 = 0. 
From the above calculations, we see that the manifold under consideration satisfies n(€) = 


—1 and Vxé = ¢X. Hence the structure (¢,£,7, g) is a LP-Sasakian manifold. 


Using (3.6), we find V, the quarter-symmetric metric connection on M following: 


Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter-Symmetric Metric Connection 55 


and 
Ves€1 _ 0, Veq€2 = 0, Ve3€3 =0. 


Using (1.2), the torson tensor T, with respect to quarter-symmetric metric connection V as 
follows: 


T (e:, €:) = 0, Vi= 1, 2,3, 
T(e1, €2) =0, T(e1,e3) = e3, T(e2, e3) = e2. 


Also, 


(Veig)(€2, €3) = 0, (Veog)(€s; e1) = 0, (Vesg)(€1, e2) = 0. 


Thus M is LP-Sasakian manifold with quarter-symmetric metric connection V. 


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56 Santu Dey and Arindam Bhattacharyya 


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Math. Combin. Book Ser. Vol.1(2016), 57-64 


On Net-Regular Signed Graphs 


Nutan G.Nayak 


Department of Mathematics and Statistics 


S. S. Dempo College of Commerce and Economics, Goa, India 


E-mail: nayaknutan@yahoo.com 


Abstract: A signed graph is an ordered pair © = (G,o), where G = (V, E) is the under- 
lying graph of © and a0: E — {+1, —1}, called signing function from the edge set E(G) of 
G into the set {+1,—1}. It is said to be homogeneous if its edges are all positive or nega- 
tive otherwise it is heterogeneous. Signed graph is balanced if all of its cycles are balanced 


otherwise unbalanced. It is said to be net-regular of degree k if all its vertices have same 





net-degree k ie. k = d§(v) = d$(v) — ds(v), where d$(v)(d5(v)) is the number of posi- 





tive(negative) edges incident with a vertex v. In this paper, we obtained the characterization 
of net-regular signed graphs and also established the spectrum for one class of heterogeneous 


unbalanced net-regular signed complete graphs. 


Key Words: Smarandachely k-signed graph, net-regular signed graph,co-regular signed 
graphs, signed complete graphs. 


AMS(2010): 05C22, 05C50. 


§1. Introduction 


We consider graph G is a simple undirected graph without loops and multiple edges with n 
vertices and m edges. A Smarandachely k-signed graph is defined as an ordered pair © = (G,o), 
where G = (V, F) is an underlying graph of © and o: E > {@7,@,@3,--- ,@%} is a function, 
where @ € {+,—}. A Smarandachely 2-signed graph is known as signed graph. It is said to be 
homogeneous if its edges are all positive or negative otherwise it is heterogeneous. We denote 
positive and negative homogeneous signed graphs as +G and —G respectively. 

The adjacency matrix of a signed graph is the square matrix A(X) = (a;;) where (i,j) 
entry is +1 if o(u,v;) = +1 and —1 if o(v;v;) = —1, 0 otherwise. The characteristic polynomial 
of the signed graph © is defined as ®(© : A) = det(AJ — A(X)), where J is an identity matrix 
of order n. The roots of the characteristic equation ®(% : A) = 0, denoted by Aq, A2,-++ An 
are called the eigenvalues of signed graph X. If the distinct eigenvalues of A(X) are A, > 
Ag > +++ > Xp» and their multiplicities are m1, me2,...,7n, then the spectrum of } is Sp(X) = 
FOS A, Sok oe 


Two signed graphs are cospectral if they have the same spectrum. The spectral criterion 


lReceived May 19, 2015, Accepted February 15, 2016. 


58 Nutan G.Nayak 


for balance in signed graph is given by B.D.Acharya as follows: 


Theorem 1.1([1]) A signed graph is balanced if and only if it is cospectral with the underlying 
graph. i.e. Sp(XS) = Sp(G). 


The sign of a cycle in a signed graph is the product of the signs of its edges. Thus a cycle 
is positive if and only if it contains an even number of negative edges. A signed graph is said 
to be balanced (or cycle balanced) if all of its cycles are positive otherwise unbalanced. The 
negation of a signed graph © = (G,o), denoted by n(X) = (G,o) is the same graph with all 
signs reversed. The adjacency matrices are related by A(—X) = —A(X). 


Theorem 1.2((12]) Two signed graphs X11 = (G,o1) and Xp = (G,a2) on the same underlying 
graph are switching equivalent if and only if they are cycle isomorphic. 


In signed graph 5, the degree of a vertex v is defined as sdeg(v) = d(v) = d§(v) + d5(v), 
where d$(v)(ds(v)) is the number of positive(negative) edges incident with v. The net degree 





of a vertex v of a signed graph © is d=(v) = d$(v) —ds(v). It is said to be net-regular of degree 





k if all its vertices have same net-degree equal to k. Hence net-regularity of a signed graph can 
be either positive, negative or zero. We denote net-regular signed graphs as U*. We know [13] 
that if 4 is a & net-regular signed graph, then & is an eigenvalue of © with j as an eigenvector 
with all 1’s. 

K.S.Hameed and K.A.Germina [6] defined co-regularity pair of signed graphs as follows: 


Definition 1.3({6]) A signed graph © = (G,c) is said to be co-regular, if the underlying graph 
G is regular for some positive integer r and & is net-regular with net-degree k for some integer 


k, and the co-regularity pair is an ordered pair of (r,k). 


The following results give the spectra of signed paths and signed cycles respectively. 


Lemma 1.4((3]) The signed paths PO, where r is the number of negative edges andO<r< 


n—1, have the eigenvalues(independent of r) given by 


Td . 
Aj = 2 —,j=1,2,---,n. 
J Foe te Tey »n 


Lemma 1.5([9]) The eigenvalues A; of signed cycles CO” and0<r<n are given by 


(23 = [rx 


Aj = 2cos 
where r is the number of negative edges and [r] = 0 if r is even, [r] = 1 ifr ts odd. 


Spectra of graphs is well documented in [2] and signed graphs is discussed in [3, 4, 5, 9]. 
For standard terminology and notations in graph theory we follow D.B.West [10] and for signed 
graphs T. Zaslavsky [14]. 

The main aim of this paper is to characterize net-regular signed graphs and also to prove 


On Net-Regular Signed Graphs 59 


that there exists a net-regular signed graph on every regular graph but the converse does 
not hold good. Further, we construct a family of connected net-regular signed graphs whose 
underlying graphs are not regular. We established the spectrum for one class of heterogeneous 
unbalanced net-regular signed complete graphs. 


§2. Main Results 


Spectral properties of regular graphs are well known in graph theory. 


Theorem 2.1([2]) If G is anr regular graph, then its maximum adjacency eigenvalue is equal 
tor andr = 2m 

Here we generalize Theorem 2.1 to signed graphs as graph is considered as one case in 
signed graph theory. We denote total number of positive and negative edges of © as m* and 


m~— respectively. The following lemma gives the structural characterization of signed graph 
so that © is net-regular. 


Lemma 2.2 If © = (G,c) is a connected net-regular signed graph with net degree k then 
k= 2M where M = (mt —m7), m* is the total number of positive edges and m~ is the total 
number of negative edges in %. 


Proof Let © = (G,o) be a net-regular signed graph with net degree k. Then by definition, 
d=(v) = d£(v) — dS (v). Hence, 


> 45 (e) = ase) — Dds (0). 








Thus, 
nk = > d(v) — 45 (0). 
Whence, 7 
k= = Ee Ea ‘s(n = = [2m* — 2m") 


i=1 w=1 
2(mt —m-) ae 2M 











I 





Corollary 2.3 If =(G,o) is a signed graph with co-regularity pair (r,k) then r > k. 


Proof Let & be a k net-regular signed graph then by Lemma 2.2, k = =", where M = 


n? 


—m). Since G is its underlying graph with regularity r on n vertices then r = 2 ,where 


n? 
‘ae 
2n me) Hence r > k. 


(m+ 


m=m*t+m_. It is clear that 2m > 














Remark 2.4 By Corollary 2.3, if © = (G,o) is a signed graph with co-regularity pair (r,k) on 


60 Nutan G.Nayak 


n vertices then —r <k <r. 


Now the question arises whether all regular graphs can be net-regular and vice-versa. From 
Lemma 2.2, it is evident that at least two net-regular signed graphs exist on every regular graph 
when m+ = 0 or m~ =0. We feel the converse also holds good. But contrary to the intuition, 
the answer is negative. Next result proves that underlying graph of all net-regular signed graphs 
need not be regular. 


Theorem 2.5 Let ¥ be a net-regular signed graph then its underlying graph is not necessarily 


a regular graph. 


Proof Let & be a net-regular signed graph with net degree k. Then by Lemma 2.2, 





k= ona), By changing negative edges into positive edges we get k = * where 
m=mt+m-. Ifk= 2m is a positive integer then underlying graph is of order k = r. If 


k= 2mm is not a positive integer then k £ r. Hence the underlying graph of a net-regular signed 











graph need not be a regular graph. 





Shahul Hameed et.al. [7] gave an example of a connected signed graph on n = 5 whose 
underlying is not a regular graph. Here we construct an infinite family of net-regular signed 
graphs whose underlying graphs are not regular. 


Example 2.6 Here is an infinite family of net-regular signed graphs with the property that 
whose underlying graphs are not regular. Take two copies of C’,, join at one vertex and assign 
positive and negative signs so that degree of the vertex common to both cycles will have net 
degree 0 and also assign positive and negative signs to other edges in order to get net-degree 0. 


The resultant signed graph is a net-regular signed graph with net-degree 0 whose underlying 


(0) 
(2n—1) 


and 3. In chemistry, underlying graphs of these signed graphs are known as spiro compounds. 


graph is not regular. We denote it as © for each C,, and illustration is shown in Fig.1, 2 


In the following figures, solid lines represent positive edges and dotted lines represent 
negative edges respectively. 








Fig.1 Net-regular signed graph D2 for C3 


On Net-Regular Signed Graphs 61 


Fig.2 Net-regular signed graph 9 for C4 


\ 
\ 
\ 
\ 


Fig.3 Net-regular signed graph ©} for Cs 


From Figures 1, 2 and 3, we can see that =? is a bipartite signed graph, but U2 and D3 
are non-bipartite signed graphs. The spectrum of these net -regular signed graphs are 


























Sp(X2)= {+2.2361, +1, 0}, 
Sp(=2)= {+2.4495, +1.4142, (0)°}, 
Sp(X3)={+2.3028, +1.6180, +1.3028, +0.6180, 0}. 


0) 
2n—1) 
property i.e. spectrum is symmetric about the origin and also these are non-bipartite when 


Remark 2.7 Spectrum of this family of connected signed graphs x satisfy the pairing 


cycle C,, is odd. 


Heterogeneous signed complete graphs which are cycle isomorphic to the underlying graph 
+K,, will have the spectrum {(n — 1),(—1)("~)} and which are cycle isomorphic to —Ky, 
will have the spectrum {(1— 7), (1)(~)}. Here we established the spectrum for one class of 
heterogeneous unbalanced net-regular signed complete graphs. 


Let C,, be a cycle on n vertices and C,, be its complement where n > 4. Define o : 


62 Nutan G.Nayak 


E(Kn) a {1,-1} by 
1, ifeECy 


a(e) = = 
-1, ifeeC, 


Then © = (Ky, ¢) is an unbalanced net-regular signed complete graph and we denote it as K"** 
where n > 4. 


The following spectrum for kK” is given by the author in [10]. 


Lemma 2.8({10]) Let kK” be a heterogeneous unbalanced net-regular signed complete graph 

then 

5—n 1+ 4cos(724) 
1 1 


Sp( Kn) = 


where (5 —n) gives the net-regularity of Kk". 


Lemma 2.9 w™ +w""" = 2cos 2L forl<j<nand1<r<n, where w is the n'® root of 


unity. 


Proof Letl<j<nandil<r<n. 




















_ Qrnijg 2 —lQrrig Qrrig —2Qrnrij 
w” +w Te saa + e27T @ = =e +e nm 
2rmj. , rm 2rmj. . arm 
= cos + 74sin —— + cos —7sin 
n n n 
2rmj 
= 2cos ‘ 
n 











By using the properties of the permutation matrix [8] and from Lemma 2.9, we give a new 
spectrum for kt. 


Theorem 2.10 Let K"* be a heterogeneous signed complete graph as defined above. If n is 
odd then 





n-1 
277 
Sp(Kn*) = {2 —-_5 m) 
‘D(A ) = {2.cos n 2u cos 





2rmj ce ea. 
n 
and if n is even then 
nee 
net 20j : - 2rm7 : 
Spl kp ) = {2cos —= — cosmj — © 2cos :1<j<n}. 
Q r=2 


Proof Label the vertices of a circulant graph as 0,1,---,(m— 1). Then the adjacency 


On Net-Regular Signed Graphs 63 


matrix A is 


0 Cl C2 Cn—1 
nme 0 C1 Cuts 
A= Cn—-2 Cn-1 0 Cn-3 , 
C1 C2 C3 0 


where ¢; = Cn_; = 0 if vertices i and n —7 are not adjacent and c; = cyn_; = 1 if vertices 7 and 


n — 1% are adjacent. 


Hence 
A = cP) +c9P? +---+¢n_1P"™ + 
n-1 
= doe’, 
r=1 


where P is a permutation matrix. 


Let K” be the heterogeneous signed complete graph and A(K”*) be its adjacency matrix. 





A(K") is a circulant matrix with first row [0,1,—1,—1,--- ,—1,1]. Here cy = 1,cg = —1,¢3 





—1,-++ ,Cn-1 =1. Hence A(K"**) can be written as a linear combination of permutation matrix 
P. A(Knet) = pl — p? — p3...— pr-2 4 pr, 





Case 1. If 7n is odd then 





A(KR*) = {(P! + PP) — (P24 pr)... (P*F + P*H)| 


and w € Sp(P). Hence 

















Sp(Knt) = {(wl+u"))- Ww? tw") —--- wt +0" )} 
Qn 2(2= 1) nj 
= {200s 4 —... eos EI} 
n n 
Raa 
Pe 205 a 2rmy . 
Spike) = 2cos = — S © 2cos :l<j<n 
i. ey n 
Case 2. If n is even then 
A(Knt) = {(P} + P*}) — (P? + Pt?) —...— (P*F* + p*#) — (P#)} 
and w € Sp(P). Hence 
Sp(Knt) = {wt +w"}) - +0") —--- WF +w"F) - W)} 
; na 2(2)107 
= {200s 28 — 2005 i) 658 (3) sinh 
nm nm 


64 Nutan G.Nayak 


So 


n-2 


9) . e208 
Sp(Kn) = ¢ 2cos aes cos 17} — ye 2. cos 
n 
r=2 


2rm7 














:l<jg<n 





Acknowledgement The author thanks the University Grants Commission(India) for provid- 
ing grants under minor research project No.47-902/14 during XII plan. 


References 


1] B.D.Acharya, Spectral criterion for cycle balance in networks, J. Graph Theory, 4(1980), 
1-11. 

2] D.M.Cvetkovic, M.Doob, H.Sachs, Spectra of Graphs, Academic Press, New York, 1980. 

3] K.A.Germina, K.S.Hameed, On signed paths, signed cycles and their energies, Applied 
Math Sci., 70(2010) 3455-3466. 

4) K.A.Germina, K.S.Hameed, T.Zaslavsky, On product and line graphs of signed graphs, 
their eigenvalues and energy, Linear Algebra Appl., 435(2011) 2432-2450. 

5] M.K.Gill, B.D.Acharya, A recurrence formula for computing the characteristic polynomial 
of a sigraph, J. Combin. Inform. Syst. Sci., 5(1)(1980) 68 - 72. 

6] K.S.Hameed, K.A.Germina, On composition of signed graphs, Discussiones Mathematicae 
Graph Theory, 32(2012) 507-516. 

7| K.S.Hameed, V.Paul, K.A.Germina, On co-regular signed graphs, Australasian Journal of 
Combinatorics, 62(2015) 8-17. 

8] R.A.Horn, C.R.Johnson, Matriz Analysis, Cambridge University Press, Cambridge, 1985. 
9| A.M.Mathai, T.Zaslavsky, On Adjacency matrices of simple signed cyclic connected graphs, 
J. of Combinatorics, Information and System Sciences, 37(2012) 369-382. 

[10] N.G.Nayak, Equienergetic net-regular signed graphs, International Journal of Contempo- 
rary Mathematical Sciences, 9(2014) 685-693. 

[11] D.B.West, Introduction to Graph Theory, Prentice-Hall of India Pvt. Ltd., 1996. 

[12] T.Zaslavsky, Signed graphs, Discrete Appl.Math., 4(1982) 47-74. 

[13] T.Zaslavsky, Matrices in the theory of signed simple graphs, Advances in Discrete Math- 








ematics and Applications, (Ramanujan Math. Soc. Lect. Notes Mysore, India), 13(2010) 
207-229. 
T.Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, 
(Manuscript prepared with Marge Pratt), Journal of Combinatorics, DS, NO.8(2012), 
pp.1-340. 


= 
fiat 


Math.Combin. Book Ser. Vol.1(2016), 65-75 


On Common Fixed Point Theorems 


With Rational Expressions in Cone l-Metric Spaces 


G.S.Saluja 


Department of Mathematics, Govt. Nagarjuna P.G. College of Science 
Raipur - 492010 (C.G.), India 


E-mail: salujal963@gmail.com 


Abstract: In this paper, we establish some common fixed point theorems for rational 
contraction in the setting of cone b-metric spaces with normal solid cone. Also, as an 
application of our result, we obtain some results of integral type for such mappings. Our 


results extend and generalize several known results from the existing literature. 
Key Words: Common fixed point, rational expression, cone b- metric space, normal cone. 


AMS(2010): 47H10, 54H25. 


§1. Introduction and Preliminaries 


Fixed point theory plays a very significant role in the development of nonlinear analysis. In 
this area, the first important result was proved by Banach in 1922 for contraction mapping in 
complete metric space, known as the Banach contraction principle [2]. 

In 1989, Bakhtin [3] introduced b-metric spaces as a generalization of metric spaces. He 
proved the contraction mapping principle in b-metric spaces that generalized the famous con- 
traction principle in metric spaces. Czerwik used the concept of b-metric space and generalized 
the renowned Banach fixed point theorem in b-metric spaces (see, [5, 6]). In 2007, Huang and 
Zhang [9] introduced the concept of cone metric spaces as a generalization of metric spaces and 
establish some fixed point theorems for contractive mappings in normal cone metric spaces. In 
2008, Rezapour and Hamlbarani [14] omitted the assumption of normality in cone metric space, 
which is a milestone in developing fixed point theory in cone metric space. 

In 2011, Hussain and Shah [10] introduced the concept of cone b-metric space as a general- 
ization of b-metric space and cone metric spaces. They established some topological properties 
in such spaces and improved some recent results about kK KM mappings in the setting of a cone 
b-metric space. 

In this note, we establish some common fixed point theorems satisfying rational inequality 


in the framework of cone b-metric spaces with normal solid cone. 


Definition 1.1({9]) Let E be a real Banach space. A subset P of E is called a cone whenever 


1Received August 12, 2015, Accepted February 16, 2016. 


66 G.S.Saluja 


the following conditions hold: 


(C1) P is closed, nonempty and P 4 {0}; 

(C2) a,bE R, a,b>0 and x,y € P imply ax + by € P; 

(C3) PN (—P) = {0}. 

Given a cone P C E, we define a partial ordering < with respect to P by x < y if and only 
ify—x EP. We shall write x < y to indicate thatx < y butx Ay, while x < y will stand for 
y—2x € P°, where P° stands for the interior of P. If P’ 40 then P is called a solid cone (see 


[15]). 


There exist two kinds of cones- normal (with the normal constant A) and non-normal ones 
following ((7]): 


Let E be a real Banach space, P C Ea cone and < partial ordering defined by P. Then 
P is called normal if there is a number K > 0 such that for all x,y € P, 


O<e2<y imply [ell < Koll, (1.1) 
or equivalently, if (Vn) an < Yn < Zn and 


lim ¢, = lim z,=2 imply lim y, =. (1.2) 
n— Co noo N—- CoO 


The least positive number K satisfying (1.1) is called the normal constant of P. 


Example 1.2 ([15]) Let E = Cj(0, 1] with ||z|| = ||z||,, + ||2’||, on P = {x € E: x(t) > O}. 


This cone is not normal. Consider, for example, x,,(t) = ae and y,(t) = 4, Then 0 < tp < Yn, 
and limn—oo Yn = 0, but |[¢n|| = maxze(o,1] \-| + maxye(o,1] |t"-1| = + +1 > 1; hence xp, does 


not converge to zero. It follows by (1.2) that P is a non-normal cone. 


Definition 1.3((9, 16]) Let X be a nonempty set. Suppose that the mapping d: X x X > E 


satisfies: 


(CM1) 0< d(az,y) for allz,ye X witha #y and d(a,y) =0 © r=y; 
(CM2) d(x,y) =d(y,x) for all x,y € X; 
(CM3) d(x, y) < d(x, z) +d(z,y) vy,z EX. 


Then d is called a cone metric on X and (X,d) is called a cone metric space (CMS). 


The concept of a cone metric space is more general than that of a metric space, because 
each metric space is a cone metric space where & = R and P = (0, +00). 


Example 1.4 ([9]) Let FE = R?, P={(z,y) € R?: 2>0,y> 0}, X =Randd: XxX 5E 
defined by d(x, y) = (|x — y|,a|a — y|), where a > 0 is a constant. Then (X,d) is a cone metric 
space with normal cone P where K = 1. 


Example 1.5 ((13]) Let B = 0?, P= {{2n}n>1 € E: ny > 0, for all n}, (X,p) a metric space, 


and d: X x X — E defined by d(z, y) = {p(a, y)/2” }n>1. Then (X,d) is a cone metric space. 


On Common Fixed Point Theorems with Rational Expressions in Cone b-Metric Spaces 67 


Clearly, the above examples show that class of cone metric spaces contains the class of 


metric spaces. 


Definition 1.6([10]) Let X be a nonempty set and s > 1 be a given real number. A mapping 
d: X x X — E is said to be cone b-metric if and only if, for all x, y, z € X, the following 


conditions are satisfied: 


(CbM1) 0 < d(a,y) witha #y and d(a,y) =0 © «=y; 
(CbM2) d(x, y) = d(y, 2); 

(CbM3) d(a, y) < s[d(a, z) + d(z, y)]. 

The pair (X,d) is called a cone b-metric space (CbMS). 


Remark 1.7 The class of cone b-metric spaces is larger than the class of cone metric space 
since any cone metric space must be a cone b-metric space. Therefore, it is obvious that cone 
b-metric spaces generalize b-metric spaces and cone metric spaces. 


We give some examples, which show that introducing a cone b-metric space instead of a 
cone metric space is meaningful since there exist cone b-metric spaces which are not cone metric 


spaces. 


Example 1.8 ([8]) Let E = R*, P = {(z,y) € E: x2 >0,y > 0} Cc FE, X = R and 
d: X x X — E defined by d(x,y) = (|% — y|?, ala — y|?), where a > 0 and p > 1 are two 
constants. Then (X,d) is a cone b-metric space with the coefficient s = 2? > 1, but not a cone 
metric space. 


Example 1.9 ((8]) Let X = @? with 0 < p < 1, where @ = {{r,} CR: 07), |an|? < cof. 
Let d: X x X — R, defined by 


Co 


a(e,y) = (So len”)? 


n=1 


where « = {rn}, y = {yn} © @. Then (X,d) is a cone b-metric space with the coefficient 
s = 2? > 1, but not a cone metric space. 
Example 1.10 ([8]) Let X = {1,2,3,4}, FE = R?, P= {(z,y) € E: x >0,y > 0}. Define 
d: X x X — E by 
(Ie—yl~*,|e—yl"") if Ay, 
0, ife=y. 


d(x, y) _ 


Then (X, d) is a cone b-metric space with the coefficient s = $ > 1. But it is not a cone metric 
space since the triangle inequality is not satisfied, 


d(1,2) > d(1,4) +.4(4,2), (3,4) > d(3, 1) +d(1,4). 


Definition 1.11({10]) Let (X,d) be a cone b-metric space, x € X and {xy} be a sequence in 
X. Then 


68 G.S.Saluja 


e {x,} is a Cauchy sequence whenever, if for every c € E with 0 < c, then there is a 
natural number N such that for alln,m > N, d(tn,2m) < c; 

e {x} converges to x whenever, for everyc € E with0 <c, then there is a natural number 
N such that for alln > N, d(an,x) < c. We denote this by limp.otn = L Or Lyn > XU as 
n— oo. 


e (X,d) is a complete cone b-metric space if every Cauchy sequence is convergent. 


In the following (X,d) will stands for a cone b-metric space with respect to a cone P with 
P° £0) in areal Banach space E and < is partial ordering in E with respect to P. 


§2. Main Results 


In this section we shall prove some common fixed point theorems for rational contraction in the 


framework of cone b-metric spaces with normal solid cone. 


Theorem 2.1 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient s > 1 
and P be a normal cone with normal constant kK. Suppose that the mappings S,T: X — X 


satisfy the rational contraction: 


d(x, Sx) d(x, Ty) + [d(z, y)|? + d(x, Sx)d(a, y) 


U(Sx,Ty) < al d(x, Sx) + d(x,y) + d(x, Ty) 


(2.1) 





for all x,y € X, a € (0,1) with sa < 1 and d(x, Sx) + d(z,y) + d(x,Ty) # 0. Then S and 
T have a common fixed point in X. Further if d(x, Sx) + d(a,y) + d(x,Ty) = 0 implies that 
d(Sx,Ty) =0, then S and T have a unique common fixed point in X. 


Proof Choose 2p € X. Let #1 = S(ao) and x2 = T(#1) such that von41 = S(xan) and 
Lon42 = T(€2n41) for alln > 0. Let d(a, Sx) + d(x, y) + d(a,Ty) £0. From (2.1), we have 


A(®an41, L2n42) = A(Sx2n, T%2n+41) 
al (dean, Strom) d(xan, Tan41) + [d(xen, Z2n+1)]? 


IA 


+d(X2n, S£2n) d(Lan, an+1)) 
=i 
x (d(e2n, Sxon) + d(%an,®an41) + d(xan, Tan41)) 
= al (der, Lon41) U(Lan, Lan+2) + [d(ren, Lan+1)]° 
+d(Xan, Van41) d(Zan, tan+1)) 


-1 
x d( Z2Qn; Lon41) +d(Xan, Lon41) Pde d(t2n, 2n42)) 








(Lon, Lon42) + d(Xan, Lon41) +r d(L2n, ant) 
d(t2n, Lon41) +d(Xan, Lon41) as d(L2n, Lon+2) 
(an, L2n41)- (2.2) 


x 





(1 

= a a Lon41) 
a 
d 


= a 


On Common Fixed Point Theorems with Rational Expressions in Cone b-Metric Spaces 69 


Similarly, we have 


d(tan, Ton+41) = A(Sx2n,T%2n—1) 
a [ (dan; Stan) d(Lan, Tt2n—1) + [d(van, Ln-1)|° 


+d(Zon, Son) d(Lon, 2n-1)) 


IA 


=a 
x (d(e2n, Sion) +d Pons Put) + dein; Ti2n-1)) 

= al (den, Lon41) Aan, Lan) + [d(xen, T2n—-1)]* 
+d(Zon, Gan+1) d(Lan, an-1)) 


. (a Lan, Zan+1) + d(ton, an—1) + d(xen, tn) | 


= ad(Lan,X2n-1) 
E Lan, L2n-1) aCe all 

d(Xan,Lan41) + d(@an, Lan—1) 

= ad(Xan,Xan-1). (2.3) 


x 





By induction, we have 


A(tn41, De) <a Gee a) < a d(tn—2, Bact) <... 
<a” d(x, 21). (2.4) 


Let m,n > 1 and m > n, we have 


d(Ln,Lm) 8{d(an,2n41) + A(@n41,Lm)| 


= sd(Xn,Xn41) + $d(@n41, 0m) 


11) + $7[d(tn41,2n42) + d(tn42,2m)| 


A 
& 
Q 
8 
os 
8 
; 


In; En41) + $°d(tn41,2n42) + $°d(an42,2m) 





IA 
% 
Q 
8 
= 
8 
7 


11) + 87d(2n41, 2n42) + 8°d(tn+42, En+3) 


























giiiaas |C erenmae ee Lm) 
< sa"d(x1,29) + 82a" d(a1, 29) + 88a"? d (x1, 20) 
s™att™—1 d(x1, £0) 
= sa[l+sa+s?o? + s%a° +---+(sa)™ *]d(21, 20) 
sa” 
< d . 
> ea (71,0) 


Since P is a normal cone with normal constant K, so we get 





l|d(an,2m)|| < K— 


< Ke |d(e1, 20). 


This implies ||d(a,, %m)|| > 0 as n,m — oo since 0 < sa < 1. Hence {2,,} is a Cauchy sequence. 
Since (X, d) is a complete cone b-metric space, there exists z € X such that x, > z as n — ov. 


70 G.S.Saluja 


Now, since 
d(z,Tz) <  s{d(z,2en41) + d(ran41,Tz)] 
= sd(Sxon,Tz) + sd(z, t2n+41) 
2 pc S2on) d(ton, Tz) + [d(xen, z)|? + d(ren, Sx2n)d(ren, =) 
os d(Lan, SLan) + d(on, z) + d(tan, Tz) 
+sd(z,®2n+41) 
o[ Ln+1) (Lan, Tz) + [d(xan, 2)|? + d(ran, F2n41)d(xan, 2) 
= A(Lon,L2n41) + (Lan, Z) + d(an, Tz) 


+sd(z,@2n+41). 
Now using the condition of normal cone, we have 


d(X2n,2n41) U(tan, Tz) + [d(r2n, 2)? + d(an, F2n41)d(@2n, 2) 
< pe Nr i a A AL i ae ar a nh 
Mates Pep 2 K{ sal | d(Xan,Lan41) + d(ten, 2) + d(Lan, Tz) 


+s||d(z, ton). 
As n — 00, we have 
lIa(z,T2)| < 0. 


Hence ||d(z, Tz)|| = 0. Thus we get Tz = z, that is, z is a fixed point of T. 

In an exactly the same fashion we can prove that Sz = z. Hence Sz = Tz = z. This shows 
that z is a common fixed point of S and T. 

For the uniqueness of z, let us suppose that d(x, Sax) + d(x,y) + d(z,Ty) = 0 implies 
d(Sx,Ty) = 0 and let w be another fixed point of S and T in X such that z 4 w. Then 


d(z,Sz)+d(z,w)+d(z,Tw) =0 => d(Sz,Tw) =0. 


Therefore, we get 
d(z,w) = d(Sz,Tw) =0, 


which implies that z = w. This shows that z is the unique common fixed point of S and T. 











This completes the proof. 





If S is a map which has a fixed point p, then p is a fixed point of $” for every n € N too. 
However, the converse need not to be true. Jeong and Rhoades [12] discussed the situation and 
gave examples for metric spaces, while Abbas and Rhoades [1] examined this for cone metric 
spaces. If a map satisfies F'(S') = F(S”) for each n € N then it is said to have property P. If 
F(S")O F(£") = F(S)N F(T) then we say that S and T have property P*. 

We examine the property P* for those mappings which satisfy inequality (2.1). 


Theorem 2.2 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient s > 1 
and P be a normal cone with normal constant kK. Suppose that the mappings S,T: X — X 
satisfy (2.1). Then S and T have the property P*. 


On Common Fixed Point Theorems with Rational Expressions in Cone b-Metric Spaces 71 


Proof By the above theorem, we know that S and T have a common fixed in X. Let 
Then 


z € F(S")N F(T"). 


d(z,Tz) 


Similarly 


dis zr a) 


IA 


dso £2) Sa TT) 
al (a(srtz, Sz) d(S"-1z, T(T"2z)) + [d(S"-4z, Tz)? 


4d(S"-12z, Sz) d(S"-1z, Tz) 

x (d(S"-1z, 5"2) + d(S*-12, Tz) + a(s"12, T(T"2))) | 
al (a(s"tz, z)d(S"-1z, Tz) + [d(S"-*z, z)]? 

+d(S"-1z, z) d(S"-1z, 2)) 


x (a S"-1z, 2) +d(S"-1z, z) +d(S"-1z Te) ‘ 
d 


d(S"—1z, Tz) + 2d(S" +z, 2) 


n-1 
OE? 218) Pepommeenercare 


ads? az), 


IA 


ads” ait" 2) Sass" Ar 2) 
al (a(s"-?z, S™1z) d(8"-2z, Tz) + [d(S"-22z, T"-12)) 


IA 


4d(S"-2z, 8"-1z) d(S"-22, pet) 
-1 
x(a (Sn-2z, gn} 2) + a(S"-7z, T°-12) + a(S"-?z, T"2)) 
= al (a d(S"-2z, §"-1z) d(S"-2z,T™z) + [d(S"-2z, 8°12)? 
4d(S"-2z, 8°12) d(S"-2z, s*-12)) 


-1 
x (a (S"-22, 9"-1z) 4 d(S"-2z, S"-1z) + d(S"-2z, TZ) 


d(S* 72.7%) ob a(S" 22, aH 


= n— 2, n—-1 oS 
= ad(S lee 2B Poe Sr-1z) + d(S"-2z,T%z) 


= od(S* 72,5" 12). 


Continuing this process, we get that 


d(S"z,T" tz 


That is, 


) 


dS oT) ar aS! a TOs) as So eT), 


d(z,Tz) <a" d(z,Tz). 


Using (1.1), the above inequality implies that 


|d(z,Tz)|| < Ka” ||d(z,Tz)|| ~ 0asn— ow. 


(2 G.S.Saluja 


Hence ||d(z, Tz)|| = 0. Thus we get Tz = z, that is, z is a fixed point of T. By using Theorem 











2.1, we get Sz = z, and consequently, S and T have property P*. This completes the proof. 





Putting S = T, we have the following result. 


Corollary 2.3 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient 
s > 1 and P be a normal cone with normal constant K. Suppose that the mappings T: X — X 


satisfies the rational contraction: 


d(x,Tx) d(x, Ty) + [d(2, y)|? + d(x, Tx)d(a, y) 


d(Tx,Ty) < al d(x,Tx) + d(x,y) + d(x, Ty) 


(2.5) 





for all z,y € X, a € (0,1) with sa < 1 and d(x,Tx) + d(z,y) + d(x,Ty) 4 0. Then T has a 
fixed point in X. Further if d(x,Tx) + d(x, y) + d(x,Ty) = 0 implies that d(Tx,Ty) = 0, then 
T has a unique fixed point in X. 


Proof The proof of Corollary 2.3 immediately follows from Theorem 2.1 by taking S = T. 











This completes the proof. 





Theorem 2.4 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient s > 1 
and P be a normal cone with normal constant K. Suppose that the mapping T: X — X satisfies 
(2.5) with sa <1, where a € [0,1). Then T has the property P. 


Proof Let v € F(T”). Then 


d(v,Tv) = d(T"v,T"tv) =d(T(T™ ‘v), T(T"v)) 

a arntv, T"v) d(T"-1v,T(T"v)) + [d(T"-1v, Tv)? 
+d(T""!v, T"v) d(T" 1v, Tv) 

x {d(T"-'v, Tv) + d(T"-!v, T"v) + d(T""v, T(T"~))} | 


IA 


- afar, v) d(T"-1v, Tv) + [d(T"-1, v)/? 
+d(T”1v, v)d(T”*v, v) 
x {d(T"~*v,v) +d(L"1v,v) + d(T", T)} | 


d(T”—1'v, Tv) + 2d(T”~1v, v) 


= peat CaN BS Le SS 
oe Uaehes Paes v) + d(T"—!v, Tv) 


= ad(T”1v,v). 
That is 
d(T"v,T**1v) < ad(T"'v, Tv). 


Similarly 


On Common Fixed Point Theorems with Rational Expressions in Cone b-Metric Spaces 73 


d(T”—'v,T"v) 


I 


d(T (I ?v), T(T” *v)) 

ofa(r"-2v, T"-1v) d(T"-2v, T"v) + [a(T"-2v, T"'v)? 
+d(T"-*u, Tv) d(T” 70, T” 1v) 

x {d(T"-7v,T"*v) + d(I"-7, T” *v) + d(T” 70, Tv) }] 


IA 


= ald(T"-?v,T"-'v) a(T"“V0, v) + [a(T"-2v, T™10)P 
+d(T"-?u,T”1u) d(T” *0, T” |v) 
x {d(T"-7v,T"*v) + d(T" 70, T”*v) + d(T”? 0, ye] 


d(T”—1v, v) + 2d(T"—20, | 


= d Tr-2 qTr-l 
oa - Us 2d(T"—2u, T"—1v) + d(T” 10, v) 


ad(T” *v,T" 'v). 


l| 


Continuing this process, we get 


d(T, Tv) < ad(T"~1v,T"v) < a? d(T” 20, T™ 1v) < --- < a" d(v, Tv). 


That is, 
d(v,Tv) <a” d(v, Tv). 


Using (1.1), the above inequality implies that 
|d(v,Tv)|| < Ka” ||d(v,Tv)|| - 0asn— oo. 


Hence ||d(v, T'v)|| = 0. Thus we get Tv = v. Thus we conclude that a mapping which satisfies 














(2.5) has the property P. This completes the proof. 


§3. Applications 


The aim of this section is to apply our result to mappings involving contraction of integral 
type. For this purpose, denote A the set of functions y: [0,00) — [0, co) satisfying the following 
hypothesis: 


(h1) y is a Lebesgue-integrable mapping on each compact subset of [0, 00); 


(h2) for any e > 0 we have f) y(t) dt > 0. 


Theorem 3.1 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient s > 1 
and P be a normal cone with normal constant kK. Suppose that the mappings S,T: X — X 


74 G.S.Saluja 


satisfy the contraction of integral type: 


pr wwe < af 


for all x,y € X, a € [0,1) with sa<1landWeA. Then S and T have a unique common fixed 





| stearate tue? tate sede 
d(x,Sx)+d(za, x,Ty 


v(t) dt 


point in X. 


If we put S = T in Theorem 3.1, we have the following result. 


Theorem 3.2 Let (X,d) be a complete cone b-metric space (CCbMS) with the coefficient s > 1 
and P be a normal cone with normal constant K. Suppose that the mapping T: X — X satisfies 


the contraction of integral type: 


d(«,Tx) d(«,Ty)+[d(a,y)]*+d(«,Tx)d(a,y) 


a(Tx,Ty) Ta Ta) tae y) ae, To) 
if w(t)dt < a | w(t) dt 
0 0 


for allz,y€ X, a€ (0,1) with sa<landWweA. Then T has a unique fixed point in X. 








§4. Conclusion 


In this paper, we establish some unique common fixed point theorems for rational contraction 
in the setting of cone b-metric spaces with normal solid cone. Also, as an application of our 
result, we obtained some results of integral type for such mappings. Our results extend and 
generalize several results from the existing literature. 


References 


[1] M.Abbas and B.E.Rhoades, Fixed and periodic point results in cone metric spaces, Appl. 
Math. Lett., 22(4)(2009), 511-515. 

[2] S.Banach, Surles operation dans les ensembles abstraits et leur application aux equation 
integrals, Fund. Math., 3(1922), 133-181. 

[3] I.A.Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. 
Gos. Ped. Inst. Unianowsk, 30(1989), 26-37. 

[4] S-H Cho and J-S Bae, Common fixed point theorems for mappings satisfying property 
(E.A) on cone metric space, Math. Comput. Model., 53(2011), 945-951. 

[5] S.Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostraviensis, 
1(1993), 5-11. 

[6] S.Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti. Semin. 
Mat. Fis. Univ. Modena, 46(1998), 263-276. 

[7] K.Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. 

[8] H.Huang and S.Xu, Fixed point theorems of contractive mappings in cone b-metric spaces 
and applications, Fixed Point Theory Appl., 112(2013). 


10 


11 


12 


13 





(14 
[15 


[16 





On Common Fixed Point Theorems with Rational Expressions in Cone b-Metric Spaces 75 


L.-G.Huang and X.Zhang, Cone metric spaces and fixed point theorems of contractive 
mappings, J. Math. Anal. Appl., 332(2)(2007), 1468-1476. 

N.Hussain and MH. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl., 
62(2011), 1677-1684. 

S.Jankovié, Z.Kadelburg and S.Radenovicé, On cone metric spaces: a survey, Nonlinear 
Anal., 4(7)(2011), 2591-2601. 

G.S.Jeong and B.E.Rhoades, Maps for which F(T) = F(T"), Fixed Point Theory Appl., 
6(2004), 71-105. 

Sh.Rezapour, A review on topological properties of cone metric spaces, in Proceedings of 
the International Conference on Analysis, Topology and Appl. (ATA 08), Vrinjacka Banja, 
Serbia, May-June 2008. 

Sh.Rezapour and R.Hamlbarani, Some notes on the paper “Cone metric spaces and fixed 
point theorems of contractive mappings”, J. Math. Anal. Appl., 345(2)(2008), 719-724. 
J.Vandergraft, Newton method for convex operators in partially ordered spaces, SIAM J. 
Numer. Anal., 4(1967), 406-432. 

P.P.Zabrejko, K-metric and K-normed linear spaces: survey, Collectanea Mathematica, 
48(4-6)(1997), 825-859. 


Math. Combin. Book Ser. Vol.1(2016), 76-81 


Binding Number of Some Special Classes of Trees 


B.Chaluvaraju, H.S.Boregowda? and S.Kumbinarsaiah? 
1Department of Mathematics, Bangalore University, Janana Bharathi Campus, Bangalore-560 056, India 
2 Department of Studies and Research in Mathematics, Tumkur University, Tumkur-572 103, India 


Department of Mathematics, Karnatak University, Dharwad-580 003, India 
E-mail: bchaluvaraju@gmail.com, bgsamarasa@gmail.com, kumbinarasaiah@gmail.com 


Abstract: The binding number of a graph G = (V, E£) is defined to be the minimum of 
|N(X)|/|X| taken over all nonempty set X C V(G) such that N(X) # V(G). In this article, 


we explore the properties and bounds on binding number of some special classes of trees. 
Key Words: Graph, tree, realizing set, binding number, Smarandachely binding number. 


AMS(2010): 05C05. 


§1. Introduction 


In this article, we consider finite, undirected, simple and connected graphs G = (V, EF) with 
vertex set V and edge set E. As such n =| V | and m =| E | denote the number of vertices 
and edges of a graph G, respectively. An edge - induced subgraph is a subset of the edges of a 
graph G together with any vertices that are their endpoints. In general, we use (X) to denote 
the subgraph induced by the set of edges X C E. A graph G is connected if it has a u — v 
path whenever u,v € V(G) (otherwise, G is disconnected). The open neighborhood of a vertex 
v € V(G) is N(v) = {u € V : wv € E(G)} and the closed neighborhood N[v] = N(v) U {v}. 
The degree of v, denoted by deg(v), is the cardinality of its open neighborhood. A vertex with 
degree one in a graph G is called pendant or a leaf or an end-vertex, and its neighbor is called 
its support or cut vertex. An edge incident to a leaf in a graph G is called a pendant edge. 
A graph with no cycle is acyclic. A tree T is a connected acyclic graph. Unless mentioned 
otherwise, for terminology and notation the reader may refer Harary [3]. 

Woodall [7] defined the binding number of G as follows: If X C V(G), then the open 
neighborhood of the set X is defined as N(X) = U,ex N(v). The binding number of G, 
denoted 6(G), is given by 
IN(X)| 

|X| ¢ 
where F = {X CV(G):X 4 9,N(X) 4 V(G)}. We say that b(G) is realized on a set X if 
X € Fand 0(G) = aot , and the set X is called a realizing set for b(G). Generally, for a given 
graph H, a Smarandachely binding number bx (G) is the minimum number 6(G) on such F with 


b(G) = Minzer 





1Received July 23, 2015, Accepted February 18, 2016. 


Binding Number of Some Special Classes of Trees 77 


(X)¢ # A for VX € F. Clearly, if H is not a spanning subgraph of G, then by(G) = b(G). 
For complete review and the following existing results on the binding number and its related 
concepts, we follow [1], [2], [5] and [6]. 


Theorem 1.1 For any path P, with n > 2 vertices, 


1 if n is even; 


b(Pr) = 


3 


1 5 ae 
Feel if n is odd. 


Theorem 1.2 For any spanning subgraph H of a graph G, b(G) < b(#). 


In [8], Wayne Goddard established several bounds including ones linking the binding num- 
ber of a tree to the distribution of its end-vertices end(G) = {v € V(G) : deg(v) = 1}. Also, let 
o(v) = |N(v) Nend(G)| and o(G)= max {o(v) : v € V(G)}. The following result is obviously 
true if e(G) = 0 and if o(G) = 1, follows from taking X = {N(v)Mend(G)}, where v is a vertex 
for which e(v) = e(G). 


Theorem 1.3 For any graph G, 0(G).b(G) < 1. 


Theorem 1.4 For any nontrivial tree T, 


(1) 02’) = 1/A(T); 
(2) 0(T) 2 1/o(T) +1. 


§2. Main Results 


Observation 2.1 Let T be a tree with n > 3 vertices, having (n — 1)-pendant vertices, which 
are connected to unique vertex. Then b(T) is the reciprocal of number of vertices connected to 


unique vertex. 
Observation 2.2 Let T be a nontrivial tree. Then b(T) > 0. 


Observation 2.3 Let T be a tree with b(T) < 1. Then every realizing set of T is independent. 


Theorem 2.4 For any Star Ky -1 with n > 2 vertices, 





1 
b(Kin-1) = : 
n-1 
Proof Let Ky,n-1 be astar with n > 2 vertices. If Ky,,-1 has {v1, v2,-+- , Un} vertices with 
deg(v1) = n—1 and deg(v2) = deg(v3) = --- = deg(vn) = 1. We prove the result by induction 


on n. For n = 2, then |N(X)| = |X| = 1 and (Ky) = 1. For n = 3, |N(X)| < |X| = 2 and 
b(K4,2) = 4. Let us assume the result is true for n = k for some k, where k is a positive integer. 


Hence b(/y,,-1) = ms 


78 B.Chaluvaraju, H.S.Boregowda and S.Kumbinarsaiah 


Now we shall show that the result is true for n > k. Since (& + 1)- pendant vertices in 
Ki x41 are connected to the unique vertex v;. Here newly added vertex vz+1 must be adjacent 
to v, only. Otherwise Ky,,41 loses its star criteria and vz,+1 is not adjacent to {v2,U3,°-+ , UK}, 
then Ky 441 has k number of pendant vertices connected to vertex v;. Therefore by Observation 
2.1, the desired result follows. 














Theorem 2.5 Let T, and T, be two stars with order n, and nz , respectively. Then ny < ng 
if and only if b(T) > b(Z). 


Proof By Observation 2.1 and Theorem 2.4, we have b(T;) = + and 0(T2) = et Due to 
the fact of ny < ng if and only if + > ms Thus the result follows. 














Definition 2.6 The double star Ky, is a tree with diameter 3 and central vertices of degree r 
and s respectively, where the diameter of graph is the length of the shortest path between the 
most distanced vertices. 

Theorem 2.7 For any double star Kx, with 1 <r< s vertices, 


r,s 


1 


Me) nasty 1 


Proof Suppose Ky, is a double star with 1 <r < s vertices. Then there exist exactly 
two central vertices a and y for all x,y € V(K7.,) such that the degree of x and y are r and s 
respectively. By definition, the double star K;, is a tree with diameter 3 having only one edge 
between x and y. Therefore the vertex x is adjacent to (r — 1)-pendant vertices and the vertex 
y is adjacent to (s — 1)-pendant vertices. 


Clearly max{r — 1,s — 1} pendant vertices are adjacent to a unique vertex x or y as the 











case may be. Therefore b(K7. ,) = mastroLeciy: Hence the result follows. 





Definition 2.8 A subdivided star, denoted Kj,,_, 1s a star K1n-1 whose edges are subdivided 


once, that is each edge is replaced by a path of length 2 by adding a vertex of degree 2. 


Observation 2.9 Let Ky,,-1 be a star with n > 2 vertices. Then cardinality of the vertex set 
of Kj,,-1 isp=2n—1. 


Theorem 2.10 For any subdivided star K{,,_, with n = 2 vertices, 


4 ifn = 2; 
b( KY n-1) ai 2 ifn = 3; 
1 otherwise. 


Proof By Observation 2.9, the subdivided star Aj ,,_, has p= 2n— 1 vertices. Then the 


following cases arise: 


Binding Number of Some Special Classes of Trees 79 


Case 1. Ifn = 2, then by Theorem 1.1, b(K7 4 1) = b(P3) = 


wl vir 


Case 2. Ifn = 3, then by Theorem 1.1, b(K7 3 1) = b(Ps) = 


Case 3. If a vertex vy; € V(Ki 1) with deg(v1) = n— 1 and deg(N(v1)) = 1, where 
N(v1) = {v2,03,°+: ,Un}. Clearly, each edge {v1 v2, v1U3,-++ ,U1Un} takes one vertex on each 
edge having degree 2, so that the resulting graph will be subdivided star K7.,,_1, in which {v1} 
and {v2, U3,--- , Un} vertices do not lose their properties. But the maximum degree vertex v1 is 
a cut vertex of Kj,,_,. Therefore b(K1,,-1) < b(K{,,_1) for n > 4 vertices. Since each newly 
added vertex {u;} is adjacent to exactly one pendent vertex {v;},where i = j and 2<i,j <n, 
in K{,,-1. By the definition of binding number |N(X)| = |X|. Hence the result follows. 














Definition 2.11 A B,, graph is said to be a Banana tree if the graph is obtained by connecting 
one pendant vertex of each t-copies of an k-star graph with a single root vertex that is distinct 


from all the stars. 
Theorem 2.12 For any Banana tree By, with t > 2 copies and k > 3 number of stars, 


1 
b( Bx) = pas 


Proof Let t be the number of distinct k-stars. Then it has k — 1-pendant vertices and 


the binding number of each k-stars is m— But in B;%, each ¢ copies of distinct k-stars are 


joined by single root vertex. Then the resulting graph is connected and each k-star has k — 2 
number of vertices having degree 1, which are connected to unique vertex. By Observation 2.1, 











the result follows. 





Definition 2.13 A caterpillar tree C*(T) is a tree in which removing all the pendant vertices 


and incident edges produces a path graph. 


For example, b(C*(K1)) = 0; b(C*(P2)) = b(C*(Pa)) = 1; B(C*(Ps)) = 3; B(C*(Ps)) = 2 
and b(C* (Ka n— 1)) = a. 


Theorem 2.14 For any caterpillar tree C*(T) with n > 3 vertices, 


b(Kin—1) < O(C*(T)) < (Pr). 


Proof By mathematical induction, if n = 3, then by Theorem 1.1 and Observation 2.1, 
we have b(K1,2) = b(C*(T')) = b(P3) = 4. Thus the result follows. Assume that the result is 
true for n = k. Now we shall prove the result for n > k. Let C*(T’) be a Caterpillar tree with 
k + 1-vertices. Then the following cases arise: 


Case 1. If k +1 is odd, then b(C*(T)) < <4. 


Case 2. If k+ 1 is even, then 0(C*(T)) <1 
< 


By above cases, we have b(C*(T)) < b(P, ,) Since, k vertices in C*(T) exist k-stars, which 


80 B.Chaluvaraju, H.S.Boregowda and S.Kumbinarsaiah 














contributed at least ms Hence the lower bound follows. 


Definition 2.15 The binary tree B* is a tree like structure that is rooted and in which each 


verter has at least two children and child of a vertex is designated as its left or right child. 


To prove our next result we make use of the following conditions of Binary tree B*. 


C: If B* has at least one vertex having two children and that two children has no any 
child. 


Co: If B* has no vertex having two children which are not having any child. 
Theorem 2.16 Let B* be a Binary tree with n > 3 vertices. Then 


if B* satisfy Cy; 


i 
b(B*) = 2 
b(P,) if B* satisfy Co. 


Proof Let B* be a Binary tree with n > 3 vertices. Then the following cases are arises: 


Case 1. Suppose binary tree B* has only one vertex, say v; has two children and that two 
children has no any child. Then only vertex v; has two pendant vertices and no other vertex has 
more than two pendant vertices. That is maximum at most two pendant vertices are connected 
to unique vertex. There fore b(B*) = 4 follows. 


Case 2. Suppose binary tree B* has no vertex having two free child. That is each non-pendant 
vertex having only one child, then this binary tree gives path. This implies that 6(B*) = b(P,,) 














with n > 3 vertices. Thus the result follows. 


Definition 2.17 The t-centipede Cf is the tree on 2t-vertices obtained by joining the bottoms 
of t - copies of the path graph Pp» laid in a row with edges. 


Theorem 2.18 For any t-centipede Cf with 2t-vertices, 


b(C*) =1. 


Proof Ifn = 1, then tree Cf is a 1-centipede with 2-vertices. Thus b(C}) = 1. Suppose 
the result is true for n > 1 vertices, say n = t for some t, that is b(C;}) = 1. Further, we prove 
n=t+1, b(C4,,) = 1. Ina (t+1) - centipede exactly one vertex from each of the (k+1)- copies 
of P2 are laid on a row with edges. Hence the resulting graph must be connected and each 
such vertex is connected to exactly one pendant vertex. By the definition of binding number 
|N(X)| = |X|. Hence the result follows. 














Definition 2.19 The Fire-cracker graph Fi, is a tree obtained by the concatenation of t - 


copies of s - stars by linking one pendant vertex from each. 


Binding Number of Some Special Classes of Trees 81 


Theorem 2.20 For any Fire-cracker graph F;,, with t > 2 and s > 3. 


1 
b( Fis) = =p 


Proof If s = 2, then Fire-cracker graph F;,2 is a t-centipede and b(Fi,2) = 1. If t > 2 and 
s > 3, then t - copies of s - stars are connected by adjoining one pendant vertex from each 
s-stars. This implies that the resulting graph is connected and a Fire-cracker graph F;,,. Then 
this connected graph has (s — 2)-vertices having degree 1, which are connected to unique vertex. 











Hence the result follows. 





Theorem 2.21 For any nontrivial tree T, 


1 
n—-1 





<O(T) <1. 
Further, the lower bound attains if and only if T = Ky n-1 and the upper bound attains if the 
tree T has 1-factor or there exists a realizing set X such that XN N(X) = 9. 


Proof The upper bound is proved by Woodall in [7|with the fact of 6(T) = 1. Let 
X ¢€ F and ao = 0(G). Then |N(X)| > 1, since the set X is not empty. Suppose, 
|N(X)| > n—-6(T) +1. If 6(7) = 1, then any vertex of T is adjacent to atleast one vertex 


in X. This implies that N(X) = V(T), which is a contradiction. There fore |X| < n-— 1 and 
b(T) = |N(X)|/|X| > 1/(n — 1). Thus the lower bound follows. 














Acknowledgments The authors wish to thank Prof.N.D.Soner for his help and valuable 
suggestions in the preparation of this paper. 


References 


1] I.Anderson, Binding number of a graphs: A Survey, Advances in Graph Theory, ed. V. R. 
Kulli, Vishwa International Publications, Gulbarga (1991) 1-10. 

2] W.H.Cunningham, Computing the binding number of a graph, Discrete Applied Math. 
27(1990) 283-285. 

3] F.Harary, Graph Theory, Addison Wesley, Reading Mass, (1969). 

4) V.G.Kane, S.P.Mohanty and R.S.Hales, Product graphs and binding number, Ars Combin., 
11 (1981) 201-224. 

5] P.Kwasnik and D.R.Woodall, Binding number of a graph and the existence of k- factors, 
Quarterly J. Math. 38 (1987) 221-228. 

6] N.Tokushinge, Binding number and minimum degree for k-factors, J. Graph Theory 13(1989) 
607-617. 

7| D.R.Woodall, The binding number of agraph and its Anderson number, J. Combinatorial 
Theory Ser. B, 15 (1973) 225-255. 

8] Wayne Goddard, The Binding Number of Trees and K(;\3)-free Graphs, J. Combin. Math. 
Combin. Comput, 7 (1990)193-200. 








Math. Combin. Book Ser. Vol.1(2016), 82-90 


On the Wiener Index of 


Quasi-Total Graph and Its Complement 


B.Basavanagoud and Veena R.Desai 


(Department of Mathematics Karnatak University, Dharwad - 580 003, India) 
E-mail: b.basavanagoud@gmail.com, veenardesai6f@gmail.com 


Abstract: The Wiener index of a graph G denoted by W(G) is the sum of distances 
between all (unordered) pairs of vertices of G. In practice G corresponds to what is known 
as the molecular graph of an organic compound. In this paper, we obtain the Wiener index 
of quasi-total graph and its complement for some standard class of graphs, we give bounds 
for Wiener index of quasi-total graph and its complement also establish Nordhaus-Gaddum 


type of inequality for it. 
Key Words: Wiener index, quasi-total graph, complement of quasi-total graph. 


AMS(2010): 05C12. 


§1. Introduction 


Let G be a simple, connected, undirected graph with vertex set V(G) = {v1, v2,--+ ,Un} and 
edge set E(G) = {e1,€2,--: ,€m}. The distance between two vertices v; and v;, denoted by 
d(v;,v;) is the length of the shortest path between the vertices v; and v; in G. The shortest 
v; — v; path is often called geodesic. The diameter diam(G) of a connected graph G is the 
length of any longest geodesic. The degree of a vertex v; in G is the number of edges incident 
to v; and is denoted by d; = deg(v;) [2]. 

The Wiener index (or Wiener number) [8] of a graph G denoted by W(G) is the sum of 
distances between all (unordered) pairs of vertices of G. 


W(G) =) au, 05). 
i<j 
The Wiener index W(G) of the graph G is also defined by 
1 
WGE)= 5 dL dry), 


vi ,0jeV (G) 


where the summation is over all possible pairs v;,v; € V(G). 


The Wiener polarity index [8] of a graph G denoted by Wp(G) is equal to the number of 


lReceived May 8, 2015, Accepted February 20, 2016. 


On the Wiener Index of Quasi-Total Graph and Its Complement 83 


unordered vertex pairs of distance 3 of G. In [8], Wiener used a linear formula of W(G) and 
Wp(G) to calculate the boiling points tg of the paraffins, i-e., 


tp = aW(G) + bWp(G) +c, 


where a, b and ¢ are constants for a given isomeric group. 

Line graphs, total graphs and middle graphs are widely studied transformation graphs. 
Let G = (V(G), E(G)) be a graph. The line graph L(G) [11] of G is the graph whose vertex 
set is E(G) in which two vertices are adjacent if and only if they are adjacent in G. 

The middle graph M(G) [11] of G is the graph whose vertex set is V(G) U E(G) in which 
two vertices x and y are adjacent if and only if at least one of x and y is an edge of G, and they 
are adjacent or incident in G. The quasi-total graph P(G) of a graph G is the graph whose 
vertex set is V(G) U E(G) and two vertices are adjacent if and only if they correspond to two 
nonadjacent vertices of G or to two adjacent edges of G or one is a vertex and other is an edge 
incident with it in G. This concept was introduced in [6]. The complement of G, denoted by G, 
is the graph with the same vertex set as G, but where two vertices are adjacent if and only if 
they are nonadjacent in G. We denote the complement of quasi-total graph P(G) of G by P(G). 
Its vertex set is V(G)U E(G) and two vertices are adjacent if and only if they correspond to two 
adjacent vertices of G or to two nonadjacent edges of G or one is a vertex and other is an edge 
nonincident with it in G. In [9], it is interesting to see that the transformation graph G~ tT is 
exactly the quasi-total graph P(G) of G, and Gt~~ is the complement of P(G). Many papers 
are devoted to quasi-total graphs [1, 3, 6, 9, 10]. 

In the following we denote by Cy, Pn, Sn, Wn and Ky, the cycle , the path, the star, the 
wheel and the complete graph of order n respectively. A complete bipartite graph K,,) has 
n= a+b vertices and m = ab edges. Other undefined notation and terminology can be found 
in [2]. 


The following theorem is useful for proving our main results. 


Theorem 1.1([7]) Let G be connected graph with n vertices and m edges. If diam(G) < 2, 
then W(G) = n(n—1)—™m. 


§2. Results 


Theorem 2.1 If S, is a star graph of order n, then 
W(P(Sn)) = 3n? —5n 42. 


Proof If S, is a star graph with n vertices, m edges and )> d?=(n — 1)? + (n — 1), then 
i=l 
P(S;,) has ny =n+m = 2n— 1 vertices and 


nin—-1) Il» ‘ 
icone a a, —n 


84 B.Basavanagoud and Veena R. Desai 


edges. 


In P(S;,) distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(S;,)) = 2. 


By Theorem 1.1, W(P(S;,)) = ni(m1 — 1) — m1. Hence 








W(P(Sn)) = (2n — 1)(2n — 2) — n? +n = 3n? —5n 42. 











Theorem 2.2 If K,, is a complete graph of order n, then 


n(n> +n — 2) 


W(P(K,)) = 


Proof If Ky, is a complete graph with n vertices, m edges and S> d? = n(n — 1)?, then 
i=l 





P(k,) hasny =n+m= n’ tn vertices and 


n(n—-1) ly n(n?—n) 
ap Ne 
on 2 +32, 2 


edges. 


In P(K,,) distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(K,,)) = 2. From Theorem 1.1, 




















W(P(Kn)) = ni(ny fie 1) — M41 
_ win ntn 1 n(n?—n) n(n? +n— 2) 
gg eS 


Theorem 2.3 If W,, is a wheel graph of order n, then 


W (P(W,,)) = 2(4n? — 9n +5). 


Proof If W,, is a wheel graph with n vertices, m edges and > d?=n?+7n—8, then P(W,,) 
i=1 
has ny =n+m = 3n— 2 vertices and 


n(n — 1) 53 2 2 
A gee +3n—4 
edges. 


In P(W,,) distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(W,,)) = 2. 


From Theorem 1.1, W(P(W,,)) = ni(m1 — 1) — my. Hence, 





W(P(W,)) = (8n — 2)(8n — 2-1) — (n? + 3n — 4) = 2(4n? — 9n + 5). 














85 


On the Wiener Index of Quasi-Total Graph and Its Complement 


Theorem 2.4 If Ka.» is a complete bipartite graph of ordern = a+ b, then 





WG Sie Vere 


Proof If Ka,» is a complete bipartite graph with n = a+ b vertices, m = ab edges and 
S° d; = ab(a +d), 


i=l 


then P(Ka.») has ny =n+m=a+b-+ab vertices and 





_(ntm\(n+m—-1) 1H» (@tb)(atb+ab—1) 
ee 2 PP e 2 


edges. 
In P(Ka,») distance between adjacent vertices is one and distance between nonadjacent 


vertices is two, therefore diam(P(Ka,»)) = 2. 


From Theorem 1.1, W(P(Ka.»)) = ni(mi — 1) — m4. Therefore, 








W(P(Kap)) = (a+b+4+ab)(a+b+ab—-1) 5 
(a+b+ab—1)(a+ b+ 2ab) 

















Theorem 2.5 If P, is a path of order n > 4, then 





wPC,) == 


(P,,) has 


Proof If P, is a path with n vertices, m edges and )~ d? = 4n — 6, then 
i=1 


ny =n+m = 2n-—1 vertices and 





_ (ntm\ n(n-1) 14 9 (n—1)(3n— 2) — 2(2n — 3) 
moa (19) Map Jog eaten 
edges. 
In P(P,,) distance between adjacent vertices is one and distance between nonadjacent 





vertices is two, therefore diam(P(P,,)) = 2. 





From Theorem 1.1, W(P(P,,)) = ni(n1 — 1) — my. So 





3) _ 5n?—38n—4 





(n — 1)(38n — 2) — 2(2n 











W(P(Pn)) = (2n — 1)(2n — 2) — ; 


86 B.Basavanagoud and Veena R.Desai 


Theorem 2.6 If S;, is a star of order n > 4, then 





Proof If Sp is a star with n vertices, m edges and )> d? = (n—1)?+n-—1, then P(S',) 
i=1 
has ny =n+m = 2n-—1 vertices and m, = (5) — atn=)) Dg 4 = (n—1)? edges. 


i=l 


As diam(P(S,)) = 3. Therefore W(P(S;,)) = ni(mi — 1) — mi + W,(P(S,)), where 
W,(P(S,)) is Wiener polarity index of P(S,,). Hence, 








W(P(Sn)) = (2n—1)(2n-—2)-—(n-1)?+m 
= (2n—1)(2n—-2)-(n-1)?? +n—-1=3n(n- 1). 























Theorem 2.7 If K,, is a complete graph of order n > 4, then 





n(n? + 6n? — 5n — 2) 


W(P(Kx)) = 5 


Proof If Ky, is a complete graph with n vertices, m edges and S> d? = n(n — 1)?, then 
i=l 
P(K,) hasny =n+m= 4 eae vertices and 


_ (n+m n(n—1) 1s 2 n(n? —2n? + 3n—2) 
m = ( 2 )- 2 = 2u8 = 8 





edges. 


In P(K,,) distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(K,,)) = 2. From Theorem 1.1, 























W(P(Kn)) = ni(ny—-—1)-—m 
_ wtn ee | n(n? — 2n? + 3n — 2) 
2 2 8 
_ n(n? + 6n? — 5n — 2) 
8 


Theorem 2.8 [fC is a cycle of order n > 4, then 





n(5n + 1) 


Ww (PCa) = 


Proof If C,, is a cycle with n vertices, m edges and 3 d?=4n, then P(C,,) has 
i=l 


On the Wiener Index of Quasi-Total Graph and Its Complement 87 


ny =n+m = 2n vertices and 


m, = wae marae n(3n — 5) 


edges. 
In P(C;,,) distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(C;,)) = 2. 


From Theorem 1.1, W(P(C;,)) = ni(mi — 1) — my. So, 











W(P(Cn)) = 2n(2n—1) — BOR=S) > Ber ey 











Theorem 2.9 If Ka» is a complete bipartite graph of ordern = a-+ b, then 





WPI) = tb tab YiR(a+ b+ ab) ~ ab 


Proof If Ka,» is a complete bipartite graph with n = a+ b vertices, m = ab edges and 


S° dj = ab(a +d), 


i=l 


then P(Ka.) has ny =n+m=a+b-+ab vertices and 


_f{ntm (ntm)(n+m—-1) 1<¢ 2 ab(a+b+ab—1) 


edges. 
In P(Ka,») distance between adjacent vertices is one and distance between nonadjacent 
vertices is two, therefore diam(P(Ka,)) = 2. 


By Theorem 1.1, 


W (P(Ka,)) => ni(ny = 1) — My 
= (a+b +ad)(a+b+ab—1)~ Mash rar=h) 
(a+b+ ab—1)[2(a+6-+ ab) — ab] 


5 ’ 














Theorem 2.10 If G is a connected graph of order n, then W(G) < W(P(G)). 


Proof If G is graph with n vertices and m edges then P(G) is a quasi-total graph of G 


with n+ ™ vertices and 
nn—1) 14, 
“N\A 
ae 


edges. 


88 B.Basavanagoud and Veena R. Desai 


Wiener index of graph increases when new vertices are added to the graph G. Therefore 
W(G) < W(P(G)). 














Lemma 2.11 If G is connected graph of order n, then 


A —2 
3n? —5n+2< W(P(G)) < eae) 
and the upper bound attain if G is a complete graph and lower bound attain if G is a star graph. 


Proof Let P(G) is a quasi-total graph of G with n +m vertices and 


n 


n(n—1) 1 9 
Lae, ge 
y+ 5D 


edges. 

G has maximum edges if and only if G = K,, P(G) has maximum number of vertices if 
and only if G = Kp. 

Wiener index of a graph increases when new vertices are added to the graph and P(K,) 
has maximum number of vertices compared with any other P(G). Therefore W(P(G)) < 
W (P(Ky)). 

From Theorem 2.2, W(P(K,)) = nee), Therefore 


n(n? +n — 2) 


wiP(a@) <4 


(1) 


with equality holds if and only if G = Ky. 

For any graph G has minimum edges if and only if G © T and P(G) has minimum 
number of vertices if and only if G = 7. Wiener index of a graph increases when new vertices 
are added to the graph and P(T’) has minimum number of vertices compared with any other 
P(G). Therefore W(P(T)) < W(P(G)). In the case of tree W(P(S;,)) < W(P(T)). Therefore 
W(P(Sn)) < W(P(G)). 

From Theorem 2.1, W(P(S;,)) = 3n? — 5n + 2. Hence, 


3n? — 5n +2 < W(P(G)) (2) 


with equality if and only if G = S),. 
From equations (1) and (2), we get that 





n(n? +n — 2) 
i ; 





3n? — Bn + 2< W(P(G)) < 








Lemma 2.12 For any connected graph G of order n > 4, 


5n? — 3n—4 ——, _ n(n? + 6n? — 5n — 2) 
———_ < P << 
mat <wP@)< ; | 


and the upper bound attain if G is a complete graph and lower bound attain if G is a path. 


On the Wiener Index of Quasi-Total Graph and Its Complement 89 


Proof Let G be connected graph with n > 4 vertices and m edges. Then P(G) has n +m 


vertices and 
n(n—1) Il» 
LN 


edges. P(K,,) has n +m vertices and 


ea 


edges. 


G has maximum edges if and only if G = K,, P(G) has maximum number of vertices if 


and only if G = K,. Wiener index of a graph increases when new vertices are added to the 





graph and P(K,,) has maximum number of vertices compared to any other P(G). Therefore 
W(P(G)) < W(P(K,,)). From Theorem 2.7, 








nr n? n? — on — 
W(P(K,)) = eee 
Therefore cr . 
W(P(G) < = (3) 


For any connected graph G with n > 4 vertices, G has minimum number of vertices 
if and only if G = T. Wiener index of a graph increases when new vertices are added to 


a graph and P(T) has minimum number of vertices compared to any other P(G). Thus, 


W(P(T)) < W(P(G)). 


In case of tree W(P(P,)) < W(P(TL)). Therefore W(P(P,)) < W(P(G)). By Theorem 
2.5, W(P(Pn)) = Busey Therefore 


on ant < w(PE). (4) 


From equations (3) and (4), we get that 





5n? — 3n—4 ——.. _ n(n? + 6n? — 5n — 2) 
—————_ < W(P(G)) < —————__. 
snot <wP@)< : 











The following theorem gives the Nordhaus-Gaddum type inequality for Wiener index of 
quasi-total graph. 


Theorem 2.13 For any graph G with n > 4, 


mun 13) <W(P(G)) + W(PIG) < 3n(n? + a —n- 2) 


90 


B.Basavanagoud and Veena R. Desai 


Proof From Lemmas 2.11 and 2.12, we have 


2_ a ———— 
5n 3n—4 2 














3n? —5n +24 5 < W(P(G))+W(P(G)) 
nttn?—2n | t+ Gn3 — Sn —2n 
, 4 8 
Thus, 
nin = 18) < wpa) +w(P@) < Me tee 
Acknowledgement 


The first author on this research is supported by UGC-MRP, New Delhi, India: F. No. 41- 
784/2012 dated: 17-07-2012 and the second author on this research is supported by UGC- 
National Fellowship (NF) New Delhi. No. F./2014-15/NFO-2014-15-OBC-KAR-25873/(SA- 
III/Website) Dated: March-2015. 


References 


1 








B.Basavanagoud, Quasi-total graphs with crossing numbers, Journal of Discrete Mathe- 
matical Sciences and Cryptography, 1 (1998), 133-142. 

F.Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1969). 

V.R.Kulli, B.Basavanagoud, Traversability and planarity of quasi-total graphs, Bull. Cal. 
Math. Soc., 94 (1) (2002), 1-6. 

Li Zhang, Baoyindureng Wu, The Nordhaus-Gaddum-type inequalities for some chemical 
indices, MATCH comm, Math., Comp. Chem., 54 (2005), 189-194. 

H.S.Ramane, D.S.Revankar, A.B.Ganagi, On the Wiener index of graph, J. Indones. Math. 
Soc., 18 (1) (2012), 57-66. 

D.V.S.Sastry, B.Syam Prasad Raju, Graph equations for line graphs, total graphs, middle 
graphs and quasi-total graphs, Discrete Mathematics, 48 (1984), 113-119. 

H.B.Walikar, V.S.Shigehalli, H.S-Ramane, Bounds on the Wiener index of a graph, MATCH 
comm, Math., Comp. Chem., 50 (2004), 117-132. 

H.Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 
(1947), 17-20. 

B.Wu, J.Meng, Basic properties of total transformation graphs, J. Math. Study, 34 (2) 
(2001), 109-116. 

B.Wu, L.Zhang, Z.Zhang, The transformation graph G*¥* when ryz = — ++, Discrete 
Mathematics, 296 (2005), 263-270. 

Xinhui An, Baoyindureng Wu, Hamiltonicity of complements of middle graphs, Discrete 
Mathematics, 307 (2007), 1178-1184. 

Xinhui An, Baoyindureng Wu, The Wiener index of the kth power of a graph, Applied 
Mathematics Letters, 21 (2008), 436-440. 


Math.Combin.Book Ser. Vol.1(2016), 91-96 


Clique Partition of Transformation Graphs 


Chandrakala S.B 


(Nitte Meenakshi Institute of Technology, Bangalore, India) 


K.Manjula 


(Bangalore Institute of Technology, Bangalore, India) 


E-mail: chandrakalasb@yahoo.co.in, manju-chandru2005@rediffmail.com 


Abstract: A clique in a graph G is a complete subgraph of G. A clique partition of G is a 
collection C of cliques such that each edge of G occurs in exactly one clique in C’. The clique 
partition number cp(G) is the minimum size of a clique partition of G. In this paper upper 
bounds for the clique partition number of the transformation graphs G**~ and Gt** for 


some standard class of graphs is obtained. 
Key Words: Transformation graph, clique, clique partition. 


AMS(2010): 05C70, 05C75. 


§1. Introduction 


All graphs G considered here are finite, undirected and simple. We refer to [1] for unexplained 
terminology and notations. In 2001 Wu and Meng introduced some new graphical transfor- 
mations which generalizes the concept of the total graph. As is the case with the total graph, 
these generalizations referred to as transformation graphs G*¥* have V(G)U E(G) as the vertex 
set. The adjacency of two of its vertices is determined by adjacency and incidence nature of 
the corresponding elements in G. 

Let a, 3 be two elements of V(G) U E(G). Then associativity of a and (@ is taken as + if 
they are adjacent or incident in G, otherwise —. Let xyz be a 3-permutation of the set {+, —}. 
The pair a and 3 is said to correspond to « or y or z of xyz if a and G are both in V(G) or both 
are in £(G), or one is in V(G) and the other is in E(G) respectively. Thus the transformation 
graph G*¥* of G is the graph whose vertex set is V(G) U E(G) and two of its vertices a and 8 
are adjacent if and only if their associativity in G is consistent with the corresponding element 
of ryz. 

In particular G**— and Gt** are defined as: 


Definition 1.1 The transformation graph G**— of G is the graph with verter set V(G)U E(G) 
in which the vertices u and v are joined by an edge if one of the following holds 


1Received July 12, 2015, Accepted February 21, 2016. 


92 Chandrakala S.B and K.Manjula 


(1) both u,v € V(G) and u and v are adjacent in G; 

(1) both u,v € E(G) and u and v are adjacent in G; 

(3) one is in V(G) and the other is in E(G) and they are not incident with each other in 
G. 


Definition 1.2 The transformation graph Gt** (total graph) of G is the graph with vertex set 
V(G) U E(G) in which the vertices u and v are joined by an edge if one of the following holds 


(1) both u,v € V(G) and u and v are adjacent in G; 
(2) both u,v € E(G) and u and v are adjacent in G; 
(3) one is in V(G) and the other is in E(G) and they are incident with each other in G. 


The transformation graphs are investigated in [2], [3] and [4]. 

For convenience, the transformation graph G** is partitioned into G*¥* = S,(G)US,(G)U 
S.(G) where 5;(G), Sy(G) and S,(G) are the edge-induced subgraphs of G’*¥*. The edge set 
of each of which is respectively determined by x, y and z of the permutation xyz. S,(G) = G 
when x is + and S,(G) & G when z is —. S,(G) & L(G) when y is + and S,(G) = L(G) when 
y is —. When z is +, a, 3 € V(G*4*) are adjacent in S,(G) if they are incident with each other 
in G. When z is —, a, are adjacent in S,(G) if they are not incident in G. 

A clique partition of G is a collection C of cliques such that each edge of G occurs in 
exactly one clique in C. The clique partition number cp(G) is the minimum size of a clique 
partition of G. 

In this paper the upper bounds for clique partition number of transformation graphs Gtt— 
and Gtt* of some class of graphs such as path, cycle, star, wheel, etc, are obtained. 


§2. Clique Partition of P't~ and Ct*— 


We note that the size of P**~ and Ct+~ are n? —n—1 and n? respectively; the clique 
numbers of P*+*~ and C,t*~ is 4. Therefore no clique partition of P/*~ and Cy +~ can 
contain K;y(t > 5). 


Theorem 2.1 For a path P, (n> 8), cp(P**~) < n? —6n+7. 


Proof Consider the path P,, : vy — vg — v3 — +++ — Un. Let e; = ujvigi(1 <i <n-—1) be the 
edges of P,. The edge set of P**~ is partitioned into Ky, K3 and K’s. Vertex sets of K4’s 
and K3’s are listed as elements of the sets B;. 

When n = 0(mod 4), 


By= {{Ui, Vi41, Ei+2, €:43} t= 1, 3,5, Cee) a -< Ty; 

Bo = {{vi, Vi41, €i-3, e2}:4= 5 +1, Dig eae n— 3, nT}, 
B3 = {{vi, Vi4+1 €(2+1+44)> ecm 4ari)} t= 2,4,6, he Lo Pe, 4}, 

Ba = {{v, Vi4+1) €G—#-2), eG—2—1} t= ime gt yt 25, 


Bs = {{v(a42), U(a+3), €1, €2}, {Ua—2), U(B—1), En—2,en-1}, {Y(2), UB41),e1}}- 
When n = 2(mod 4), 








Clique Partition of Transformation Graphs 93 


By, = {{vi, viti, cite, Cita} +7 =1,3,5,---, 5 — 2}, 
Bo = {{ui, Vi4t1, Ci-3; ea} :i=n-1, n—3, n—5, aoa ,5 — 2}, 
Bs = {{vi, viti, e(a 4a), CCR +itay} 21 = 2,4,6,---, 3 — 3}, 





iow 

By= {{v(2 44); Wa i+1)> €i-1, ei} t= 3,5, 7, roe oo _ 2}, 

Bs = {{uja-1 » UB), 1, ea}, {v(a41); U(R+2), En-2, En 1}, {u2), U(B41); e3}} 
When n = 1,3(mod 4), 











By = {{vi, Viti, Cita, eiza} t= 1,3,5---,n— 2}, 
Bg = 103; Vi+1, €i-6, eio8 t= n— 3, n— 5, n— q, oN , 12, 10, 8}, 
Bs = {{un-1, Uns en 55 en 4h, {un 25 Un 1; en 7) en st, {ve, U7, €1, eo}, 


{ve, U3, En—3; €n—2}, {v4, U5, eit} 








In each case there are n — 2 K4’s and one K3. These cover all the edges of Sz, Sy and 
4n — 6 edges of S,. The remaining (n? — 7n + 8) edges of S$, are covered by Ko’s. 
Therefore P+*~ = (n— 2)K4U K3 U (n? — 7n + 8) Ka and cp(P}t~) < n? —6n +7. 














Theorem 2.2 For a cycle Cy, (n> 8), cp(O,tt~) < n? —5n. 


Proof Consider the cycle C;, : vy — vg — v3 — +++ — Un — U1. Let e; = ujviga (1 <i <n-1) 





and €n = Unv1 be the edges of C,. Edge set of C;**~ is partitioned into K4’s and K’s. Vertex 
sets of K4’s are listed as elements of the sets B; as follows: 


When n is even, 


By = {{vi, Vici, ej, ex}: for each i = 1,3,5,---,n—-3, n—1, fj =1+2(modn) andk= 
i+ 3(mod n)}, 
By = {{v;, vj, ex, ex}} for each i = 2,4,6,---,n—4 
5 +1+i(modn) when $ is odd 


k= 2 and l= k+1(mod n) 


%+i(modn) when $ is even 





, n-—2,7 =i+1(mod n)} with 


When n is odd, 





By = {{vi, Vigi, Cit2, ej}: for each i =1,3,5,---,n—4, n—2 j =14+3(mod n)}, 





Bo = {vi, Viti, €j, Ck}: for each i = 2,4,6,---,n—-—7, n—5, j =i+6(modn) k= 
i+7(mod n)}, 


Bs = {{Un, U1, €6, e7}, {Un—3, Un—2, €2, €3},{Un—1; Un; 4, est}. 


In these sets vp and eg are taken as v,, and e, respectively. 
In both the cases there are n/X4’s. These cover all the edges of S;, S, and some edges 
of S,. Remaining edges of S, are listed as K’s. Therefore C}*~ = nK4U (n? — 6n)K2 and 
cp(Ci*~) <n? —5n. 














§3. Clique Partition of Gt*~ with G isomorphic to Comb or Sunlet graphs 


The Comb graph G & P,, © Ky is the graph with path on n vertices and each vertex of path is 
adjacent to a pendant vertex. The Sunlet graph S,, = C;, © Ky is a graph with the cycle on n 


94 Chandrakala S.B and K.Manjula 


vertices and each vertex of the cycle is adjacent to a pendant vertex. 

For a comb graph G & P,, © Ky, let vi(1 < i < n) denote the vertices of P,, with v1 and vp, 
as its end vertices and e; = v;v;41 be the edges of P, and v; be the pendant vertices adjacent 
to each of v; and e}, = v;v; be the pendant edges of G. We note that order and size of V(GTT~ ) 
is 4n — 1 and 4n? — n — 3 respectively and the clique number is 5. 

For the sunlet graph S, = C, © Ki, let u;(1 < i < n) denote the vertices of C,, and v; 
be the pendant vertex adjacent to uj, e; = vivi41 be the n edges of C,, and e/ = u,v), be the 
pendant edges of $,,. We note that order and size of S;**~ is 4n and 4n? +n respectively and 
the clique number is 5. 


Theorem 3.1 Let G& P, © Ki(n > 6) be the comb graph. Then cp(G**—) < 4n? —12n +7. 


Proof Consider the comb G & P, © K,. Edge set of Gtt~ is partitioned into K;, K4, K3 
and K’s. The vertex sets of these cliques are listed as elements of sets B; are given below: 


By = {{u;, vj, Cita, Cpa} 24 = 1,2,3,--- ,n — 3}, 

Bo = {{tn—2, Un—2, Cn—-15 Cn}, {Un-1, Un—a, €1, C1}, {Uns Uns €1, €2, eat}, 

Bs = {{{ui, vigr, eg} 20 = 1,2,3,---,n-5; 7 =14+ 4}, ({ui, vga, ej} st =n—-—4n- 
3,n—2,n—1; 7 =i-—(n—5)}}. 

The sets B,, Bz and Bg cover all the edges of S,, S, and some edges of S, while remaining 
edges of S, are covered by K’s. 


Ba = {{{u;, ey}: foreach 1 <i<njl<j<nandj 4i1+2}, {{u, ef} :t=46<i< 
n—1}, {{on, eg} 52 =3 5 <t< n—-2}h, {un_a, ec}: t= 14<1< n—4}, {{on-1, ec} :4<i< 
n—3}, {{vi, ej}: for each 2 <i<n—-3;1<j<n—-landj #4i-1,1,14+1,i1+2,1+3,714+ 4}, 

Bs = {{v;, ej}: for each 1<i<n—-—3;1<j<n—landj 4i4+1,i+2}, {{vi_s, ec}: 
t=nandl<i<n—2},{{up_1, es}: 2<i<n—I]}, {{u,, es}: 3<i<n—I1}, {{uj, ef}: 
for each 1 <i<n—2;1<j <nandj 4i,i+2}, {{u,_1, ej} :t=n 2<i<n—2}, {{up, ei}: 
i= 162i <n, 


Thus, Gt*~ = (n — 2)Ks U2K4U (n — 1)K3 U (4n? — 14n + 8) Ko and hence cp(Gtt~) < 
An? —12n +7. 














Theorem 3.2 For S, ~ Cy, © Ki(n > 6) a sunlet graph, cp(S**—) < 4n? — 10n. 


Proof Consider the sunlet graph S, & C, © Ky. Edge set of $;**7 is partitioned into Ks, 
K3 and K’s where, 


Bi = {{vi, vj, ej, ex, e&}: foreach 1 <i<n, fj =i+1(modn), k=i+2(mod n)}, 

Bo = {{vi, vj, ex}: foreach 1 <i<n, j7=i+1(modn), k=i+4(mod n)}, 

Bs = {{uvi, ej}, {uj, ej} : foreach 1 <icn, 1 <j < nand j Fi,i + 2(mod n)} U 
{{vj, ej}: for eachl<i<n, 1<j<n-—landj 4i+1,i+2(mod n)}U {{vi, ej}: 
foreach 1<i<n, 1<j<n-—landj 4i-1,1,14+1,14+ 2,1+3,1+4(mod n)}. 














Thus, S**7 = nKs5 Unkz3 U (4n? — 12n) Ke and cp(St+~) < 4n? — 10n. 











Clique Partition of Transformation Graphs 95 


§4. Clique Partition of Transformation Graphs dt and Wa 


For the star graph Ky,n, let vo be the central vertex, vj(1 < i <n) be the pendant vertices and 
€; = vor; be the pendant edges. We note that |V(Kft)| = 2n+1, |E(KEF”)| = n(3n—-1)/2 
and the clique number is n. 

For the wheel graph W,,41 = C;, + Ky, let up be the central vertex, v; be the vertices, e; = 
voui(1 <4 <n) be the spokes and ef = vjvj(1 <i<n, 7 =i+1(mod n)) be the hubs of W,,41. 
Then, V(WiS) = V(Wa41) U E(Wrs), VOWS )| = 8n +1, [EWE )| = 5n(n + 1)/2 


and the clique number is n. 


Theorem 4.1 For n> 3, cp(K{f~) <n? +1. 


Proof Here S, = L(Ki,) = Kn. The clique K,, covers all the edges of Sy; S; = K1,, and 
S, =nK1»-1, which are covered by n+ n(n—1) Kbs. 

{{vo, vit} : 1 <a <n} and 

{{ui, es}: for each l<i<n, 1<j<nandj Fi} 

Therefore, Ki = K, Un? Ko and hence cp(Kyt-) <n?+1. 

















Theorem 4.2 For n> 6, cp(W,t) < 2n? —6n +1. 


Proof The edge set of W,*7 is partitioned into a K,,n K4’s, 2n K3’s and (2n? — 9n) 
Ko’s. Here, 


By, = {{e1, €2, €3, +++ ,en-1, ent}, 

Bo = {{ui, ef, e,, ex}: for each 1 <i<n, j =i+1(modn), k=i+ 2(mod n)}, 

Bz = {{vo, vi, ej}: foreach 1<i<n, j ==i+ 3(mod nj}, 

Ba = {{vi, vj, e&,}: foreach 1 <i<n, j =i+1(modn), k =i+5(mod nj}, 

Bs ={{vi, ej}: foreachl<isn, 1<j<nandj #i,i+2(modn)} U {{vi, ej}: 
foreach 1<i<n, 1<j<nandj 4i-—1,7,14+1,1+2,14+3,1+4,7+4+ 5(mod n)}. 











(In the above sets vp, €9 and ef are taken as up, e, and €,.) 














Thus (Wy) = KnUnK4U 2nK3 U (2n? — 9n)K and hence cp(W,t)) < 2n?-6n +1. 


§5. Clique Partition of Transformation Graphs 
PREY SO ty das Wat akidsie 





Theorem 5.1 For n> 3, cp(P{t*) < 2n-3. 


Proof Consider the path P, : vj — vg — v3 — +++ — Un. Let e; = v;v;41 be the edges of Py. 





We note that order, size and clique number of P}++ are 2n—1, 4n—5 and 3 respectively. The 
edges of subgraphs S, and S, are partitioned into 3s and that of S, by K}s: 


{{Vi, Vig, Cc: 1 <i<n—A1} and {{e;, e41}:1< 7 <n—-2}. 





Therefore, P**+ = (n — 1)K3U (n — 2) Ko and cp(Pitt) < 2n- 3. 











96 Chandrakala S.B and K.Manjula 


Theorem 5.2 For n> 3, cp(Cttt) < 2n. 
Theorem 5.3 For n> 3, ep(kT it *) <n+1. 
Theorem 5.4 For n > 6, cp(W,i}*) < 8n +1. 


Proof The order, size and clique number of W,*;;' are 3n + 1, (n? + 17n)/2 and n +1. 
The edge set of W,1,*;* is partitioned into a K,, 3nK4s. Here, 


By = {{é1, €2, €3, *** , E€n-1, ent}, 

By = {{ei, ej, ej}: for each 1<i<n, 7 =i—1(mod n)}, 

Bs = {{vi, vj, ej}, {vo, vi, ec} : for each 1<i<n, 7 =i+1(mod n)}. 

Here each edge of subgraphs S, and S, are present in exactly one clique of Bs and each 


edge of S, is in exactly one clique of By or Bg. Thus, Weer = K, U 3nkK3 and hence 
cp(W iT) < 8n +1. 














Theorem 5.5 Forn > 4, cp(Kj**) <n4+1. 


Acknowledgements 


The authors wish to express their gratitude to Dr.B.Sooryanarayana for his helpful comments 
and suggestions. 


References 


[1] Frank Harary, Graph Theory, Narosa Publishing House, New Delhi, 1969. 

[2] Chandrakala S.B, K.Manjula and B. Sooryanarayana, The transformation graph G*¥* when 
xyz = ++-, International Journal of Mathematical Sciences And Engineering Applica- 
tions, 3 (2009), no. I, 249-259. 

[3] Baoyindureng Wu, Li Zhang, and Zhao Zhang, The transformation graph G*¥* when xyz = 
—+4, Discrete Mathematics, 296 (2005), 263-270. 

[4] Lan Xu and Baoyindureng Wu, The transformation graph G~*—, Discrete Mathematics, 
308 (2008), 5144-5148. 


Math.Combin. Book Ser. Vol.1(2016), 97-100 


Probabilistic Bounds 


On Weak and Strong Total Domination in Graphs 


M.H.Akhbari 


(Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran) 
E-mail: mhakhbari20@gmail.com, akhbari@iauest.ac.ir 


Abstract: A set D of vertices in a graph G = (V,£) is a total dominating set if every 
vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak 
if every vertex v € V — D is adjacent to a vertex u € D such that dg(v) > dg(u). The weak 
total domination number 7~+(G) of G is the minimum cardinality of a weak total dominating 
set of G. A total dominating set D of G is said to be strong if every vertex v € V — D is 
adjacent to a vertex u € D such that dg(v) < dg(u). The strong total domination number 
yst(G) of G is the minimum cardinality of a strong total dominating set of G. We present 


probabilistic upper bounds on weak and strong total domination number of a graph. 


Key Words: Weak total domination, strong total domination, pigeonhole property, prob- 
ability. 


AMS(2010): 05069. 


§1. Introduction 


We consider finite, undirected, simple graphs. Let G be a graph, with vertex set V and edge 
set E. The open neighborhood of a vertex v € V is N(v) = {u € V | uv € E} and the 
closed neighborhood is N|v] = N(v) U {vu}. For a subset S C V, the open neighborhood is 
N(S) = Uses N(v) and the closed neighborhood is N[S] = N(S) US. If vu is a vertex of V, then 
the degree of v denoted by dg(v), is the cardinality of its open neighborhood. By A(G) = A 
and 6(G) = 6 we denote the maximum and minimum degree of a graph G, respectively. A 
subset S C V is a dominating set of G if every vertex in V — S has a neighbor in S and is a 
total dominating set (td-set) if every vertex in V has a neighbor in S. The domination number 
(G) (respectively, total domination number 7,(G)) is the minimum cardinality of a dominating 
set (respectively, total dominating set) of G. Total domination was introduced by Cockayne, 
Dawes and Hedetniemi [2]. 

In [10], Sampathkumar and Pushpa Latha have introduced the concept of weak and strong 
domination in graphs. A subset D C V is a weak dominating set (wd-set) if every vertex 
v € V—S is adjacent to a vertex u € D, where dg(v) > dg(u). The subset D is a strong 
dominating set (sd-set) if every vertex v € V —S is adjacent to a vertex u € D, where dg(u) > 


lReceived May 5, 2015, Accepted February 24, 2016. 


98 M.H.Akhbari 


dg(v). The weak (strong, respectively) domination number Yw(G) (ys(G), respectively) is the 
minimum cardinality of a wd-set (an sd-set, respectively) of G. Strong and weak domination 
have been studied for example in [4, 5, 7, 8, 9]. For more details on domination in graphs and 
its variations, see [6]. 

Chellali et al. [3] have introduced the concept of weak total domination in graphs. A 
total dominating set D of G is said to be weak if every vertex v € V — D is adjacent to a 
vertex u € D such that dg(v) > de(u). The weak total domination number Yw:(G) of G is 
the minimum cardinality of a weak total dominating set of G. The concept of strong total 
domination can be defined analogously. A total dominating set D of G is said to be strong if 
every vertex v € V — D is adjacent to a vertex u € D such that de(v) < dg(u). The strong 
total domination number yst(G) of G is the minimum cardinality of a strong total dominating 
set of G. 

We obtain probabilistic upper bounds on weak and strong total domination number of a 
graph. 


§2. Results 


We adopt the notations of [1]. Let W = W(G) be the set of all vertices v € V(G) such that 
deg(v) < deg(u) for every u € N(v). Note that W(G) may be empty, and if W(G) 4 0, then 
W(G) is independent and is contained in every weak total dominating set of G. For any vertex 
v € V(G) let deg,,(v) = {u € N(v)| deg(u) < deg(v)}. 


Theorem 2.1 Let G be a graph with V(G) — N[W] #0. If dy = min{deg,,(v)|u € V(G) — 
N|W]}, then 
6 
w(@) $217] +20 [sp (1 —"* __). 
Yen(G) $217] + 2(0— |W) (1 — 


Proof For each vertex w € W consider a vertex w’ € N(w), and let W’ = {w’|w € W}. 
Clearly |W’| < |W|. Let A be a set formed by an independent choice of vertices of G — W, 
where each vertex is selected with probability 


1 
i> \te 
Spe es a) ©, 
e (=) 


Let B C V(G) — (AU N[W)) be the set of vertices that have not a weak neighbor in A. 
Clearly E(|Al) < (n — |W]|)p. Each vertex of B has at least 6,, weak neighbors in V(G) — W. 
It is easy to show that 


Pr(v E B) — (1 — p)'+4deew (v) < (l — p)iteu | 


Thus E(|B]) < (n — |W])(1 — p)*«t!. For each a € A let a’ € N(a), and let A’ = {a'|a € A}. 
Similarly for each b € B let b’ € N(b), and let B’ = {b/|b © B}. Then clearly |A’| < |A| and 
|B’| < |B]. It is obvious that D=WUW’'UAUA'U BUB’ is a weak total dominating set for 


Probabilistic Bounds on Weak and Strong Total Domination in Graphs 99 


G. The expectation of |D]| is 





E(|D|) < 2E(\W|) + 2E(\Al) + 2E(|B)) 
< 2|W] + 2(n- |W|)p + 2(n — |W])(1 — pp)? 
< 2H] + a(n — Iw) (1- ——#__). 


(1+ dy) Fe 


By the pigeonhole property of expectation there exists a desired weak total dominating 





set. 











The proof of Theorem 2.1 implies the following upper bound, which is asymptotically same 
as the bound of Theorem 2.1. 


Corollary 2.2 Let G be a graph with V(G) — N[W] £0. If 6m = min{deg,,(v)|u € V(G) — 
N[W]}, then 


Ywr(G) < 2|W| + 2(n — |W)) a} 


Ow +1 


Proof We use the proof of Theorem 2.1. Using the inequality 1 — p < e~? we obtain that 


E(|\D|) < 2|W|+2(n—|W|)p + 2(n — |W|)(1 — py? 
2|W| + 2(n — |W|)p + 2(n — |W|)e“PO“FD 








x 


In(1 + dw) 


th 
eee en 


If we put p = 


E(|D|) < 2|W| + 2(n — |W)) (S| 


Ow +1 


By the pigeonhole property of expectation there exists a desired weak total dominating 





set. 











Next we obtain probabilistic upper bounds for strong total domination number. Let S = 
S(G) be the set of all vertices v € V(G) such that deg(v) > deg(u) for every u € N(v). Note that 
S(G) may be empty, and if S(G) 4 0, then S(G) is independent and is contained in every strong 
total dominating set of G. For any vertex v € V(G) let deg,(v) = {u € N(v)|deg(u) > deg(v)}. 
The following can be proved similar to Theorem 2.1 and Corollary 2.2, and thus we omit the 
proofs. 


Theorem 2.3 Let G be a graph with V(G)— N[S] #0. If 6, = min{deg,(v)|u € V(G)— N[S}}, 
then 


yst(G) < 2|S| + 2(n — sh (1 - aE): 


Corollary 2.4 Let G be a graph with V(G)—N[S| £0. If 6; = min{deg,(v)|u € V(G)—N[S]}, 


100 


then 


M.H.Akhbari 


yst(G) < 2|$| + 2(n — |S|) eee 


6; +1 


Acknowledgements 


This research is supported by Islamic Azad University, Estahban Branch. 


References 


1 
2 





[10 





N.Alon and J.Spencer, The Probabilistic Method, John Wiley, New York, 1992. 
E.J.Cockayne, R.M.Dawes, S.T.Hedetniemi, Total domination in graphs, Networks, 10 
(1980), 211-219. 

M.Chellali and N.Jafari Rad, Weak total domination in graphs, Utilitas Mathematica, 94 
(2014), 221-236. 

J.H.Hattingh and M.A.Henning, On strong domination in graphs, J. Combin. Math. Com- 
bin. Comput., 26 (1998), 73-92. 

J.H.Hattingh and R.C.Laskar, On weak domination in graphs, Ars Combinatoria, 49 
(1998). 

T.W.Haynes, S.T.Hedetniemi, P.J.Slater (Eds.), Fundamentals of Domination in Graphs, 
Marcel Dekker, Inc., New York, 1998. 

M.Krzywkowski, On the ratio between 2-domination and total outer-independent domina- 
tion numbers of trees, Chinese Annals of Mathematics, Series B 34 (2013), 765-776. 
D.Rautenbach, Bounds on the weak domination number. Austral. J. Combin., 18 (1998), 
245-251. 

D.Rautenbach, Bounds on the strong domination number, Discrete Math., 215 (2000), 
201-212. 

E.Sampathkumar and L.Pushpa Latha, Strong, weak domination and domination balance 
in graphs, Discrete Math., 161 (1996), 235-242. 


Math. Combin. Book Ser. Vol.1(2016), 101-108 


Quotient Cordial Labeling of Graphs 


R.Ponraj 


(Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India) 


M.Maria Adaickalam 


(Department of Economics and Statistics, Government of Tamilnadu, Chennai- 600 006, India) 


R.Kala 


(Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India) 


E-mail: ponrajmaths@gmail.com, mariaadaickalam@gmail.com, karthipyi91@yahoo.co.in 


Abstract: In this paper we introduce quotient cordial labeling of graphs. Let G be a 
(p,q) graph. Let f : V(G) — {1,2,---,p} be a1—1 map. For each edge uv assign the 
label 4 (or) | according as f(u) > f(v) or f(v) > f(u). f is called a quotient 
cordial labeling of G if |e¢(0) — e¢(1)| < 1 where e(0) and ey(1) respectively denote the 
number of edges labelled with even integers and number of edges labelled with odd integers. 
A graph with a quotient cordial labeling is called a quotient cordial graph. We investigate 


the quotient cordial labeling behavior of path, cycle, complete graph, star, bistar etc. 


Key Words: Path, cycle, complete graph, star, bistar, quotient cordial labeling, Smaran- 
dachely quotient cordial labeling. 


AMS(2010): 05C78. 


§1. Introduction 


Graphs considered here are finite and simple. Graph labeling is used in several areas of science 
and technology like coding theory, astronomy, circuit design etc. For more details refer Gallian 
[2]. The union of two graphs G and G¢ is the graph Gi UG2 with V (G1 U G2) = V (Gi )UV (G2) 
and EF (G, UG2) = E (G1) U F(G2). Cahit [1], introduced the concept of cordial labeling of 
graphs. Recently Ponraj et al. [4], introduced difference cordial labeling of graphs. Motivated 
by these labelings we introduce quotient cordial labeling of graphs. Also in this paper we 
investigate the quotient cordial labeling behavior of path, cycle, complete graph, star, bistar 
etc. In [4], Ponraj et al. investigate the quotient cordial labeling behavior of subdivided star 
S(K1,n), subdivided bistar S(Bn») and union of some star related graphs. [x] denote the 
smallest integer less than or equal to x. Terms are not defined here follows from Harary [3]. 


1Received July 9, 2015, Accepted February 25, 2016. 


102 R.Ponraj, M.Maria Adaickalam and R.Kala 


§2. Quotient Cordial Labeling 


Definition 2.1 Let G be a (p,q) graph. Let f : V(G) — {1,2,--- ,p} be an injective map. For 


each edge uv assign the label Ea (or) Lo) according as f(u) > f(v) or f(v) > flu). 


v 

f(v) fu) 

Then f is called a quotient cordial labeling of G if |ez(0) — ef(1)| < 1 where es (0) and e,(1) 
respectively denote the number of edges labelled with even integers and number of edges labelled 


with odd integers. A graph with a quotient cordial labeling is called a quotient cordial graph. 


Generally, a Smarandachely quotient cordial labeling of G respect to S C V(G) is such a 
labelling of G that it is a quotient cordial labeling on G\ S. Clearly, a quotient cordial labeling 
is a Smarandachely quotient cordial labeling of G respect to S = 0. 


A simple example of quotient cordial graph is given in Figure 1. 


1 2 


Figure 1. 


§3. Main Results 


First we investigate the quotient cordial labeling behavior of path. 


Theorem 3.1 Any path is quotient cordial. 


Proof Let P, be the path ujug---un. Assign the label 1 to uy. Then assign 2, 4,8,--- 


(< n) to the consecutive vertices until we get [4+] edges with label 0, then choose the least 


number < n that is not used as a label. That is consider the label 3. Assign the label to the 
next non labelled vertices consecutively by 3,6,12,...(< n) until we get ["5+] edges with label 
0. If not, consider the next least number < n that is not used as a label. That is choose 5. Then 
label the vertices 5, 10, 20,--- (<n) consecutively. If the total number of edges with label 0 is 
[2st n=1 


net) , then stop this process, otherwise repeat the same until we get the [44] edges with label 


0. Let S be the set of integer less than or equal to n that are not used as a label. Let t be the 
least integer such that uz is not labelled. Then assign the label to the vertices uz, ut+1,°++ , Un 


from the set S in descending order. Clearly the above vertex labeling is a quotient cordial 











labeling. 





Illustration 3.2 A quotient cordial labeling of Pi5 is given in Figure 2. 


Quotient Cordial Labeling of Graphs 103 


Figure 2 


Here, S = {5,7,9, 10, 11, 13, 14, 15} 


Corollary 3.3 [fn is odd then the cycle C, is quotient cordial. 


a quotient cordial labeling of the cycle Ch. 


Proof The quotient cordial labeling of path P,, n odd, given in Theorem 3.1 is obviously 














Next is the complete graph. 


Theorem 3.4 The complete graph K,, is quotient cordial iff in < 4. 


Proof Obviously K,,n < 4 is quotient cordial. Assume n > 4. Suppose f is a quotient 


cordial labeling of Ky. 


Case 1. nis odd. 


Consider the sets, 


oy = 


| 
so { (=H) B= ss fof) 
| 

















Sn-1 = 


2 


n+1. . n-l, — na 
integers. S_ contains integers. S'3 contains 








integers. 





Clearly, S; contains 
.., Sn—2 contains 2 integers. Each S; obviously contributes edges with label 1.T herefore 
2 


IV 


[Si] + [Sol +...+ 


er(1) Sn-1 
2 








_ eae felt 3 
- 2 2 2 


n+1 
2 














ee 


= 24+3+..0+ 








n+l 
—1 
a 


(+) SS #) | orninss) (1) 


= ie2434..4 








104 R.Ponraj, M.Maria Adaickalam and R.Kala 


Next consider the sets, 


























g! [—] [=] n—-1 

_ = n—2|?|[n—3]? UJ} ntl 
2 

g! [2] aS n—3 

Za. n—4Al’|[n—5]? vj n-1 
2 


OD 


Clearly each of the sets S/ also contributes edges with label 1. Therefore 























es(1) > [Si] + |S5|+...+ |Sn-3 
_ 7 nm val tng. “4 
~ 2 2 2 vr 
n—3 


= crear cee SS ope 


Ca. (Sa) (n= 3)(n=1) 


D =a @) 





From (1) and (2), we get 








+1)(n+3 —3)(n-1 
a) > PAMO+ |, =D 
x n?+4n+3+n?—4n+3-8 
= 8 
a 2 DET x. n(n — 1) oe 
~ 8 — A 4 : 


a contradiction to that f is a quotient cordial labeling. 


Case 2. 7 is even. 





Similar to Case 1, we get a contradiction. 











Theorem 3.5 Every graph is a subgraph of a connected quotient cordial graph. 


Proof Let G be a (p,q) graph with V(G) = {u; : 1 <7 < p}. Consider the complete graph 
kK, with vertex set V(G). Let f(u;) =7, 1 <%i<p. By Theorem 3.4, we get ef(1) > ef(0). Let 
er(1) =m-+e,(0), m € N. Consider the two copies of the star Ki. The super graph G* of G 
is obtained from Ky as follows: Take one star Ky, and identify the central vertex of the star 
with u;. Take another star Ky, and identify the central vertex of the same with uz. Let S; = 
{x : x is an even number and p < x < p+2m} and Sp = {x: « is an odd number and p < 


Quotient Cordial Labeling of Graphs 105 


x <p+2m}. Assign the label to the pendent vertices adjacent to u; from the set Siin any 
order and then assign the label to the pendent vertices adjacent to ug from the set Sg. Clearly 











this vertex labeling is a quotient cordial labeling of G*. 





Illustration 3.6 Ks is not quotient cordial but it is a subgraph of quotient cordial graph G* 
given in Figure 3. 


6 8 
1 
7 
2 
3 

9 

4 5 

Figure 3 


Theorem 3.7 Any star K1,,, is quotient cordial. 


Proof Let V(Kin) = {u,ue: 1 <i < n} and E(Ky»,) = {uu :1<%< n}. Assign the 
label 1 to the central vertex u and then assign the labels 2,3,--- ,n-+1 to the pendent vertices 











U1, U2,°** ,Un. f is a quotient cordial labeling follows from the following Table 1. 





Nature of n | ef (0) | ef (1) 


nm nm 
even aa por 
| odd : = Ss 





Table 1 
Now we investigate the complete bipartite graph Kp. 
Theorem 3.8 Ko, is quotient cordial. 


Proof Let V(K2n) = {u,v,uj:1<i< n} and E(Ko,) = {uuj,vu,:1<i< n}. Assign 
the label 1,2 respectively to the vertices u,v. Then assign the label 3,4,5,---,m-+2 to the 











remaining vertices. Clearly f is a quotient cordial labeling since e¢(0) =m+1, ef(1) =m. 





Theorem 3.9 Ky ,U KinUKin ts quotient cordial. 


Proof Let V(Kin U Kin U Kin) = {u, i, 0,0;,W,Wi:1<i< nt and (Ky, U Kin U 
Kin) = {uuy, voj,wwi:1<i<n}. Define a map f :V(Ky,UKi,U Kin) > {1,2,3,...,3n} 
by f(u) =1, f(v) = 2, f(w) = 8, 


106 R.Ponraj, M.Maria Adaickalam and R.Kala 





f (vi) = 3 + 3, l<i<n 














on 
nm is even 





Next is the bistar Bp». 


Theorem 3.10 The bistar By» 18 quotient cordial. 


Proof Let V(Bnin) = {u,Wi,v,Ui:1<i< n} and E(Bnn) = {uv, uuj,v0u, +1 <i <n}. 
Assign the label 1 to u and assign the label 2 to v. Then assign the labels 3,4,5,...,n +2 to 
the vertices u1,U2,°+: ,Un. Next assign the label n+3,n+4,...,2n+2 to the pendent vertices 


U1, U2,°°* ,Un- The edge condition is given in Table 2. 


[raweata [eo 
n= 0,1,2 (mod n) n+1] on | 





n =3 (mod n) | on [nti 


Table 3 











Hence f is a quotient cordial labeling. 





The final investigation is about the graph obtained from a triangle and three stars. 


Theorem 3.11 Let C3 be the cycle ujugugui. Let G be a graph obtained from C3 with V(G) = 
V(C3) U {0;, wi, 25 1 <i <n} and E(G) = E(Cs3) U {u1u;, uew;, u3zi 1 <i<n}. Then G is 


quotient cordial. 
Proof Define f : V(G) — {1,2,3,---,3n+3} by f(ui) =1, f(u2) = 2, f(us) = 3. 
Case 1. n=0,2,3 (mod 4). 


Define 


Quotient Cordial Labeling of Graphs 107 





f (vi) = 37+ I, l<i<n 
f(%) = 30 + 2, l<i<n 
Case 2. n=1 (mod 4). 
Define 
f (vi) = 3t + 2, l<i<n 
flw:) = 3i+1, 1Sisn 
f(%) = 31 + 3, l<i<n 





values of n e 


m=1,3 (mod 4) | 2 
rey 


Table 3 























Illustration 3.12 A quotient cordial labeling of G obtained from C3 and Kj,7 is given in 
Figure 4. 





Figure 4 


References 


[1] I.Cahit, Cordial Graphs: A weaker version of Graceful and Harmonious graphs, Ars com- 


108 








R.Ponraj, M.Maria Adaickalam and R.Kala 


bin., 23 (1987), 201-207. 

J.A.Gallian, A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 
19 (2012) #Ds6. 

F.Harary, Graph theory, Addision wesley, New Delhi (1969). 

R.Ponraj, S.Sathish Narayanan and R.Kala, Difference cordial labeling of graphs, Global 
Journal of Mathematical Sciences: Theory and Practical, 5(2013), 185-196. 

R.Ponraj and M.Maria Adaickalam, Quotient cordial labeling of some star related graphs, 
The Journal of the Indian Academy of Mathematics, 37(2)(2015), 313-324. 


Math. Combin. Book Ser. Vol.1(2016), 109-115 


Nonholonomic Frames for Finsler Space with (a,3)—Metrics 


Brijesh Kumar Tripathi!, K.B.Pandey? and R.B. Tiwari? 
1. Department of Mathematics, L.E. College, Morbi(Gujrat) 363642, India 


2. Department of Mathematics, K.N.I.T., Sultanpur,U.P., 228118, India 


3. Department of Mathematics, M.K.R.E.C.I.T., Ambedkar Nagar(U.P.), India 
E-mail: brijeshkumartripathi4@gmail.com, kunjbiharipandey05@gmail.com, tiwarirambharat@yahoo.in 


Abstract: The purpose of present paper to determine the two special Finsler spaces due 


to deformations of some special Finsler space with help of (a,3)-metrics. Consequently, we 
a? B? = a3? 
a) “a ~ (af) 


i.e. product of Matsumoto metric and Kropina metric and (II) L = (a+ 8) is =6+ Be 








obtain the non-holonomic frame for the (a,3)-metrics, such as (I) L = ( 


i.e. product of Randers metric and Kropina metric. 


Key Words: Finsler Space, (a,3)—metrics, Randers metric, Kropina metric, Matsumoto 


metric, GL-metric, Non-holonomic Finsler frame. 


AMS(2010): 53C60. 


§1. Introduction 


In 1982, P.R. Holland [1] and [2] studies a unified formalism that uses a nonholonomic frame on 
space time arising from consideration of a charged particle moving in an external electromag- 
netic field. In fact, R.S. Ingarden [3] was the first to point out that the Lorentz force law can be 
written in this case as geodesic equation on a Finsler space called Randers space. The author 
R.G. Beil [5], [6] have studied a gauge transformation viewed as a nonholonomic frame on the 
tangent bundle of a four dimensional base manifold. The geometry that follows from these 
considerations gives a unified approach to gravitation and gauge symmetries. In the present 
paper we have used the common Finsler idea to study the existence of a nonholonomic frame 
on the vertical sub bundle VTM of the tangent bundle of a base manifold M. 

In this paper, the fundamental tensor field of a Finsler space might be considered as the 
deformations of two different special Finsler spaces from the (a@,3)-metrics. Further we obtain 
corresponding frame for each of these two Finsler deformations. Consequently, a nonholonomic 
frame for a Finsler space with special (a,3)-metrics such as first is the product of Matsumoto 
metric{11] and kropina metric{11] and second is the product of Randers metric[11] and Kropina 
metric. This is an extension work of Ioan Bucataru and Radu Miron [10] and also second 
extension work of S.K. Narasimhamurthy [14]. 


Consider, a;; (x) the components of a Riemannian metric on the base manifold M, a(x, y) > 


1Received August 12, 2015, Accepted February 26, 2016. 


110 Brijesh Kumar Tripathi, K.B.Pandey and R.B. Tiwari 


0 and 0b(zx,y) > 0 Two functions on TM and B (x,y) = B; (x,y) (dx’) a vertical 1-form on 
TM. Then 


Gis (ty) = ale, aij (x) + D(x, y) Bila, y) Bi (2, y) (1.1) 


is a generalized Lagrange metric, called the Beil metric. The metric tensor g;; is also known as 
a Beil deformation of the Riemannian metric a;;. It has been studied and applied by R. Miron 
and R.K. Tavakol in General Relativity for a(x, y) = exp(20(x, y)) and b = 0. The case a(x, y) 
= 1 with various choices of b and B; was introduced and studied by R.G. Beil for constructing 
a new unified field theory [6]. 


§2. Preliminaries 


An important class of Finsler spaces is the class of Finsler spaces with(a,)—metrics [11]. The 
first Finsler spaces with (a,3)—metrics were introduced by the physicist G-Randers in 1940, are 
called Randers spaces [4]. Recently, R.G. Beil suggested a more general case and considered the 
class of Lagrange spaces with (a,)—metric, which was discussed in [12]. A unified formalism 
which uses a nonholonomic frame on space time, a sort of plastic deformation, arising from 
consideration of a charged particle moving in an external electromagnetic field in the background 
space time viewed as a strained mechanism studied by P. R. Holland [1], [2]. If we do not ask 
for the function L to be homogeneous of order two with respect to the (a,@) variables, then we 
have a Lagrange space with (a,3)—metric. Next we defined some different Finsler space with 
(a,3)—metrics. 


Definition 2.1 A Finsler space F” = (M, F(a, y)) is called with (a,3)-metric if there exists a 
2-homogeneous function L of two variables such that the Finsler metric F : TM — R is given 
by 

F*(a,y) = L(a(x,y), A(x, y)), (2.1) 
where a? (x,y) = aij(x)y’y?, a is a Riemannian metric on the manifold M, and B(x, y) = b;(x)y’ 


is a 1-form on M. 











1 (0? F? 
Consider gj; = stean the fundamental tensor of the Randers space(M,F). Taking into 
yoy 
account the homogeneity of a and F we have the following formulae: 
Os eran 
p= an ays’ P= Ajp = ayt’ 
» 1s ;; Ol . OL 
b= sy= re Ql = -=P; b;, 2.2 
EY HP aail = gal = 5 (2.2) 
; Today : . a . L 
i = Spill; = pip, = slp = Tsp = =; 
Le? ; Pp Pp L Pp x 
b;P* = B bt = # 


a: L 


Nonholonomic Frames for Finsler Space with (a, )—Metrics 111 


with respect to these notations, the metric tensors (a;;) and (g;;) are related by [13], 


L B L 
gig (@, Y) = Pa + bP; + Pb; a ania = 5 kis — Pip;) + Ll;. (2.3) 


Theorem 2.1([10]) For a Finsler space (M,F) consider the matrix with the entries: 


. a ‘ . Qa. 
Yp= (26 — 1; + \[Sr'n (2.4) 


defined on TM. Then Y; = Y}(g),5 €1,2,3,...,n is a non holonomic frame. 





Theorem 2.2([7]) With respect to frame the holonomic components of the Finsler metric tensor 


Qag ts the Randers metric gij, 1.€, 


95 = VEY os (2.5) 


Throughout this section we shall rise and lower indices only with the Riemannian metric 
ai;(x) that is ys = aizy, 8’ = a” B;, and so on. For a Finsler space with (a,3)-metric 
F? (x,y) = L(a(z, y), B(x, y)) we have the Finsler invariants [13] 


aL. Jak: ee ob. me OL pony (2.6) 
ML Seda 2 appz? P-1 = 2a Jaap’ ?-? ~ 2a2‘da2 ada : 
where subscripts 1, 0, -1, -2 gives us the degree of homogeneity of these invariants. 
For a Finsler space with metric we have, 
p18 + p_207 =0 (2.7) 


With respect to the notations we have that the metric tensor g;; of a Finsler space with 
(a,3)—-metric is given by [13] 


ij (@,Y) = parig (x) + podi(x) + p-1(bi(x) yj + bj (@)yi) + p-2yiy;- (2.8) 


From (2.8) we can see that g,;is the result of two Finsler deformations 





1 
I. ayy > hig = pai + pa 1b; + p—2yi)(p—1bj + p25) 
—2 


1 
TI. hij > Gi = hij + Fag POP a p-.1)bib;. (2.9) 


The nonholonomic Finsler frame that corresponding to the [*¢ deformation (2.9) is accord- 
ing to the Theorem 7.9.1 in [10], given by 


. A 2 ; 
Xj = V'00; — aylVe t+ yfet EG 1b' + p_2y")(p-16; + p—24;), (2.10) 
B p—2 





112 Brijesh Kumar Tripathi, K.B.Pandey and R.B. Tiwari 


where B? = aj;;(p_1b' + p_2y’)(p_1b; + p_2y;) = p20? + Bp_1p_2. 


This metric tensor a;; and h,; are related by, 





hag = XP Xjau. (2.11) 


Again the frame that corresponds to the II,q deformation (2.9) given by, 








Y; a 0% — tf (2.12) 
where C? = hijb'b! = pb? + 5+ (p-2b? + p-28)?. 
The metric tensor h;; and gi; are related by the formula 
Vn VY hg. (2.13) 


Theorem 2.3({10]) Let F?(x,y) = L(a(a,y), G(x, y)) be the metric function of a Finsler space 
with (a,8) metric for which the condition (2.7) is true. Then 


4 yiwyk 
Vj = Xi; 


is a nonholonomic Finsler frame with Xj, and Y# are given by (2.10) and (2.12) respectively. 


§3. Nonholonomic Frames for Finsler Space with (a,3)—Metrics 


In this section we consider two cases of non-holonomic Finlser frames with (a,()-metrics, such 
a I** Finsler frame product of Matusmoto metric and Kropina metric and JJ"@ Finsler frame 


product of Randers metric and Kropina metric. 

a2 B? 7 a3? 
a-B}) a a-—-B 

In the first case, for a Finsler space with the fundamental function 


_ a? (a as? 
p=(4)F- 








(I) Nonholonomic Frames for [ = ( 








the Finsler invariants (2.6) are given by 


= B _.. 1 (207 — a6?) | 
a Cr a Cr ae 


_ 1 B(G-30), __6°(a—f) 
p-1 >= 2a (a — Bs > P-2 = 203 (a — B)3’ 


_ (1 = 8a)*b? + 6°(a = B)(1 — 3a)(3a— 8) 
7 4a*(a — 6)° , 


B? = 


(3.1) 


Nonholonomic Frames for Finsler Space with (a, )—Metrics 113 


Using (3.1) in (2.10) we have, 


Peed fi ee —68 _ |4a*(a— B)® — 6430-8) 
oe Sale aa? ae By a 2a3(3a — 8) | 
i (3a— 6); —_ (3a — 8) 
x(b aa — py! 1s aa = Bye” 


Again using (3.1) in (2.12) we have, 








. 1 2(a — B)8C? \ _, 
o = 6 — a Se ] 0D; : 
Y} = 65 — ce (: 1+ 33a — 36) b'b;, (3.3) 
pe 2 B(3a—B) _ 919 2 
h 2— ___"___ ———, (a*b 
Ve 2a(a — 8)? 2a3(a — B)8 a ye 
Theorem 3.1 Consider Finsler space L = (35) x =o ak, for which the condition (2.7) is 
true. Then 
i ivk 
Vj = xi} 


is non-holomic Finsler Frame with Xj, and Y}* are given by (3.2) and (3.3) respectively. 


B 7 


(II) Nonholonomic Frames for L = (a + on \= 6? + 


In the second case, for a Finsler space with the fundamental function L = (a + p)(£) the 
Finsler invariants (2.6) are given by: 














Be : 38 +a 3B? 3 88 
Pl = ~ 5537 P0 = a oe a an a aoe? 
9 pt 
pe 4a —, (a ab? — 6°), (3.4) 
ae G2* x, -ORP can TP Oe | ges 2 OS hae 
a ors 34] 2a | Biel eines SD) 
Again using (3.4) in (2.12) we have 
i = 6 Spel tke za (3.6) 
Clee 2a+3B{ ”’ 
3 
where C? = — e 0’ 4 iy 25? — 67/7. 
203 2a 


Theorem 3.2 Consider a Finsler space L = (a + p\(£) = P+ cs for which the condition 
2.7 is true. Then 
i iyk 
Vj = x1 


is non-holomic Finsler Frame with X}, and ve are given by (3.5) and (8.6) respectively. 


114 


Brijesh Kumar Tripathi, K.B.Pandey and R.B. Tiwari 


§4. Conclusions 


Non-holonomic frame relates a semi-Riemannian metric (the Minkowski or the Lorentz metric) 


with an induced Finsler metric. Antonelli P.L., Bucataru I. ((7][8]), has been determined such 


a non-holonomic frame for two important classes of Finsler spaces that are dual in the sense of 


Randers and Kropina spaces [9]. As Randers and Kropina spaces are members of a bigger class 


of Finsler spaces, namely the Finsler spaces with(a,3)—metric, it appears a natural question: 


Does how many Finsler space with(a,()—metrics have such a nonholonomic frame? The answer 


is yes, there are many Finsler space with(a,()—metrics. 


In this work, we consider the two special Finsler metrics and we determine the non- 


holonomic Finsler frames. Each of the frames we found here induces a Finsler connection 


on TM with torsion and no curvature. But, in Finsler geometry, there are many(a,3)—metrics, 


in future work we can determine the frames for them also. 


References 


1 











Holland P.R., Electromagnetism, Particles and Anholonomy, Physics Letters, 91 (6)(1982), 
275-278. 

Holland P.R., Anholonomic deformations in the ether: a significance for the electrodynamic 
potentials. In: Hiley, B.J. Peat, F.D. (eds.), Quantum Implications, Routledge and Kegan 
Paul, London and New York, 295-311 (1987). 

Ingarden R.S., On physical interpretations of Finsler and Kawaguchi spaces, Tensor N.S., 
46(1987), 354-360. 

Randers G., On asymmetric metric in the four space of general relativity, Phys. Rev., 59 
(1941), 195-199. 

Beil R.G., Comparison of unified field theories, Tensor N.S., 56(1995), 175-183. 

Beil R.G., Equations of motion from Finsler geometric methods, In Antonelli, P.L. (ed), 
Finslerian Geometries, A meeting of minds, Kluwer Academic Publisher, FTPH, No.109(2000), 
95-111. 

Antonelli P.L., Bucataru I., On Holland’s frame for Randers space and its applications in 
physics, In: Kozma, L. (ed), Steps in Differential Geometry, Proceedings of the Colloquium 
on Differential Geometry, Debrecen, Hungary, July 25-30, 2000, Debrecen: Univ. Debrecen, 
Institute of Mathematics and Informatics, 39-54, (2001). 

Antonelli P.L., Bucataru I., Finsler connections in anholonomic geometry of a Kropina 
space, Nonlinear Studies, 8 (1)(2001), 171-184. 

Hrimiuc D., Shimada H., On the L-duality between Lagrange and Hamilton manifolds, 
Nonlinear World, 3 (1996), 613-641. 

Ioan Bucataru, Radu Miron, Finsler-Lagrange geometry applications to dynamical systems 
CEEX ET 8174/2005-2007, and CEEX M IIT 12595/2007 (2007). 

Matsumoto M., Theory of Finsler spaces with (a; 3)-metric, Rep. Math. Phys., 31, 43-83, 
(1992). 


Bucataru I., Nonholonomic frames on Finsler geometry, Balkan Journal of Geometry and 


Nonholonomic Frames for Finsler Space with (a, )—Metrics 115 


its Applications, 7 (1)(2002), 13-27. 

[13] Matsumoto M, Foundations of Finsler Geometry and Special Finsler Spaces, Kaishesha 
Press, Otsu, Japan, 1986. 

[14] Narasimhamurthy S.K., Mallikarjun Y.Kumar, Kavyashree A. R., Nonholonomic frames for 
Finsler space with special (a; 3)-metric, International Journal of Scientific and Research 
Publications, 4(1)(2014), 1-7. 


Math. Combin. Book Ser. Vol.1(2016), 116-125 


gl. 


All graphs considered in this paper are non-trivial, simple and undirected. Let G be a graph 
with vertex set V and edge set E. A k-coloring of a graph G is a partition P = {V1, V2,--- , Ve} 


of 


has a representative adjacent to at least one vertex in each of the other color classes. Such a 


coloring is called a b-coloring. The b-chromatic number was introduced by Irving and Manlove 


in 


On b-Chromatic Number of Some Line, Middle and 
Total Graph Families 


Vernold Vivin J. 


Department of Mathematics, University College of Engineering Nagercoil 


(Anna University, Constituent College) Konam, Nagercoil - 629 004, Tamilnadu, India 


Venkatachalam.M. 


Department of Mathematics, RVS Educational Trust’s Group of Institutions 


RVS Faculty of Engineering, Coimbatore - 641 402, Tamilnadu, India 


Mohanapriya N. 
Research & Development Centre, Bharathiar University, Coimbatore-641 046 and Department of 


Mathematics, RVS Technical Campus, RVS Faculty of Engineering, Coimbatore - 641 402, Tamilnadu, India 
E-mail: vernoldvivin@yahoo.in, venkatmathsQ@gmail.com, n.mohanamathsQ@gmail.com 


Abstract: A proper coloring of the graph assigns colors to the vertices, edges, or both 
so that proximal elements are assigned distinct colors. Concepts and questions of graph 
coloring arise naturally from practical problems and have found applications in many areas, 
including information theory and most notably theoretical computer science. A 6-coloring 
of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each 
color class joined to at least one vertex in each other color class. The b-chromatic number 
of a graph G, denoted by y(G), is the largest integer k such that G may have a b-coloring 
with k colors. In this paper, the authors obtain the b-chromatic number for line, middle and 


total graph of some families such as cycle, helm and gear graphs. 
Key Words: b-coloring, Helm graph, gear Graph, middle graph, total graph, line graph. 


AMS(2010): 05C15, 05C75. 


Introduction 


V into independent sets of G. The minimum cardinality k for which G has a k-coloring is 
the chromatic number y(G) of G. The b-chromatic number y(G) ([16,19,20]) of a graph G is 
the largest positive integer k such that G admits a proper k-coloring in which every color class 


[11] by considering proper colorings that are minimal with respect to a partial order defined 


1Received May 20, 2015, Accepted February 26, 2016. 


On b-Chromatic Number of Some Line, Middle and Total Graph Families 117 


on the set of all partitions of V(G). They have shown that determination of y(G) is NP-hard 
for general graphs, but polynomial for trees. There has been an increasing interest in the study 
of b-coloring since the publication of [11]. They also proved the following upper bound of y(G) 


y(G) < A(G) +1. (1.1) 


Kouider and Mahéo [12] gave some lower and upper bounds for the b-chromatic number of 
the cartesian product of two graphs. Kratochvil et al. [13] characterized bipartite graphs for 
which the lower bound on the b-chromatic number is attained and proved the NP-completeness 
of the problem to decide whether there is a dominating proper b-coloring even for connected 
bipartite graphs with k = A(G)+ 1. Effantin and Kheddouci studied in [8, 5, 6] the b- 
chromatic number for the complete caterpillars, the powers of paths, cycles, and complete 
k-ary trees. Faik [7] was interested in the continuity of the b-coloring and proved that chordal 
graphs are b-continuous. Corteel et al. [2] proved that the b-chromatic number problem is 
not approximable within 120/133 — ¢« for any « > 0, unless P = NP. Hodng and Kouider 
characterized in [10], the bipartite graphs and the Py-sparse graphs for which each induced 
subgraph H of G has y(H) = y(#). Kouider and Zaker [14] proposed some upper bounds for 
the b-chromatic number of several classes of graphs in function of other graph parameters (clique 
number, chromatic number, biclique number). Kouider and El Sahili proved in [15] by showing 
that if G is a d-regular graph with girth 5 and without cycles of length 6, then y(G) = d+ 1. 
Effantin and Kheddouci [4] proposed a discussion on relationships between this parameter and 
two other coloring parameters (the Grundy and the partial Grundy numbers). The property 
of the dominating nodes in a b-coloring is very interesting since they can communicate directly 
with each partition of the graph. 

There have been lots of works on various properties of line graphs, middle graphs and total 
graphs of graphs [1, 9, 17, 18]. 

For any integer n > 4, the wheel graph W,, is the n—vertex graph obtained by joining a 
vertex v, to each of the n — 1 vertices {w1, w2,--+ ,Wn—1} of the cycle graph C,_1. Where 
V (W,) = {v, U1, V2,°++Un—-1} and E(W,,) = {e1, e2,:++ ,en}U {51, 52,°°+ , Sn}. 

The Helm graph H,, is the graph obtained from an n-wheel graph by adjoining a pendent 
edge at each node of the cycle. 

The Gear graph G,, also known as a bipartite wheel graph, is a wheel graph with a graph 
vertex added between each pair of adjacent graph vertices of the outer cycle. 

The line graph [8] of G, denoted by L(G) is the graph with vertices are the edges of G with 
two vertices of L(G) adjacent whenever the corresponding edges of G are adjacent. 

Let G be a graph with vertex set V(G) and edge set E(G). The middle graph [8] of G, 
denoted by M(G) is defined as follows. The vertex set of M(G) is V(G) U E(G). Two vertices 
x,y in the vertex set of M(G) are adjacent in M(G) in case one of the following holds: (i) x,y 
are in E(G) and x,y are adjacent in G; (i) x is in V(G), y is in E(G), and z,y are incident in 
G. 





Let G be a graph with vertex set V(G) and edge set E(G). The total graph [8] of G, 
denoted by T(G) is defined in the following way. The vertex set of T(G) is V(G) U E(G). Two 
vertices x,y in the vertex set of T(G) are adjacent in T(G) in case one of the following holds: 


118 Vernold Vivin.J., Venkatachalam M. and Mohanapriya N. 


(i) x,y are in V(G) and « is adjacent to y in G; (#7) x,y are in E(G) and z, y are adjacent in 
G; (iti) x is in V(G),y is in E(G), and x,y are incident in G. 


§2. b-Coloring of Some Line, Middle and Total Graph Families 


Lemma 2.1 Ifn > 8 then b-chromatic number on middle graph of cycle M(C;,) is p(M(C),)) = 
5. 


Proof Let V (C,) = {v1, v2,+++ , Un} and let V (M (C),)) = {v1, v2,+++ , Un} U {u1, U2,-++ Un} 
where u; is the vertex of T (C;,) corresponding to the edge vjv;41 of C, (1 <i<n-—1). 





Fig.1 Middle Graph of Cycle M (C,,) 


Consider the following 5-coloring (ci, c2,¢3, 4, ¢5) of M (C,,) as b-chromatic: 


Assign the color c; to v1, c3 to uz, C4 to V2, C1 to U2, C5 tO U3, Co to UZ, C4 to V4, C3tO U4, 
C1 to Us, C5 tO Us, Co to Ug, C4 tO Us, C1 tO V7, C3 to U7. For 8 <i <n, assign to vertex v; one 
of the allowed colors-such color exists, because deg(v;) = 2. For 8<i<n-—1, if any, assign to 
vertex u; one of the allowed colors-such color exists, because deg(u;) = 4. An easy check shows 
that this coloring is a b-coloring. Therefore, p(M(C,,)) > 5. Since A(M(C,,)) = 4, using (1.1) 
we get that y(M(C,,)) <5. Hence, p(M(C,)) =5, Vn > 5. 














Theorem 2.2 Ifn > 5 then b-chromatic number on total graph of cycle T(C,,) is p(T (Cy)) = 5. 


Proof Let V (C;,) = {v1, v2,+++ , Un} and let V (T (C;,)) = {v1, v2,-++ Un} U {u1, U2,-+* , Un} 
where u; is the vertex of T (C;,) corresponding to the edge vjv;41 of C, (1 <i<n-—1). 


On b-Chromatic Number of Some Line, Middle and Total Graph Families 119 





Fig.2 Total Graph of Cycle T (C,,) 


Consider the following 5-coloring (ci, c2,¢3,c4,¢5) of T’(C,) as b-chromatic: assign the 
color c4 to v1, C5 to Uy, Cy tO V2, Co to U2, C3 tO V3,C4 tO UZ, C5 tO V4, Cy tO U4, Co to U5. For 
6 <i <n, assign to vertex v; one of the allowed colors-such color exists, because deg(v;) = 4. 
For 5<i<n-—1, if any, assign to vertex u; one of the allowed colors-such color exists, because 
deg(u;) = 4. An easy check shows that this coloring is a b-coloring. Therefore, y(T(C,,)) > 5. 
Since A(T(C,,)) = 4, using (1.1), we get that y(T(C,,)) <5. Hence, p(T(Cr)) =5, Vn > 5. 














Lemma 2.3 Ifn> 6 then b-chromatic number on helm graph Hy, is p(H,) = 5. 


Proof Let Hy, be the Helm graph obtained by attaching a pendant edge at each vertex of 
the cycle. Let V(H,,) = {vu} U {v1, v2,-++ , Un} U {u1, u2,-+: , Un} where v;’s are the vertices of 
cycles taken in cyclic order and u,;’s are pendant vertices such that each v;u; is a pendant edge 
and v is a hub of the cycle. 





Fig.3 Helm Graph H), 


Consider the following 5-coloring (ci, c2,¢3,¢4,¢5) of H, as b-chromatic: 


For 1 <i < 4, assign the color c; to uv; and assign the colors cs to v, C1 to U5, C3 tO Un, C4 tO 
U1, C4 tO U2, Cy to Ug, Co to ug. For 6 <i <n, assign to vertex v; one of the allowed colors-such 
color exists, because deg(v;) = 4. For 5 <i <n, if any, assign the color cq to the vertex u;. An 
easy check shows that this coloring is a b-coloring. Therefore, y(H;,) > 5. 


120 Vernold Vivin.J., Venkatachalam M. and Mohanapriya N. 


Let us assume that y(H,,) is greater than 5, ie. y(H,) = 6, V n > 6, there must be at 
least 6 vertices of degree 5 in H,,, all with distinct colors, and each adjacent to vertices of all 
of the other colors. But then these must be the vertices v, {v; : 1 < i < n}, since these are only 
ones with degree at least 4. This is the contradiction, b-coloring with 6 colors is impossible. 
Thus, we have y(H;,) < 5. Hence, y(H,) =5,V¥ n> 6. 














Lemma 2.4 If n > 7 then b-chromatic number on line graph of Helm graph L(H,) is 
p(L(Hn)) =n. 


Proof Let V(Hn) = {v} U {v1, v2,°++ , Un} U {u1, U2,-++ , Un} and E(A,) = {e;:1<i< 
n}Uf{e, :1<i<n—1}ufel }U{s; :1< i <n} where e; is the edge vu; (1 <i < n), e; is the edge 
uvigi (1 <i<n-—1), ef, is the edge v,v1 and s; is the edge vju; (1 <i <n). By the definition 
of line graph V(L(A,)) = E(An) ={e,: 1 <i< nutes: l<i<nbU{s,:1<i<n}. 





Fig.4 Line Graph of Helm Graph L(H,,) 


Consider the following n-coloring of L (H,,) as b-chromatic: For 1 <i < n, assign the color 
c, to e;. For 1 <i <n, assign to vertices s; and ej, one of the allowed colors-such color exists, 
because deg(s;) = 3 and deg(e,) = 6. An easy check shows that this coloring is a b-coloring. 
Therefore, p(L(H;,)) > n. 

Let us assume that y(L(H,,)) is greater than n, y(L(H,)) =n +1, Vn > 7, there must 
be at least n + 1 vertices of degree n in L(H,,), all with distinct colors, and each adjacent to 
vertices of all of the other colors. But then these must be the vertices e;(1 < i < n), since these 
are only ones with degree at least n. This is the contradiction, b-coloring with n+ 1 colors is 
impossible. Thus, we have y(L(H,,)) < n. Hence, y(L(H,)) =n, V n> 7. 














Theorem 2.5 Ifn > 8 then b-chromatic number on middle graph of Helm graph M(H,,) is 
p(M(Hn)) =n +1. 


On b-Chromatic Number of Some Line, Middle and Total Graph Families V0 


n}Uf{ee:1<i<n-1U {el }Uf{s,:1<%i< n} where e; is the edge vu; (1 <i <n), é is 
the edge vjvj41 (1 <i < n—1), ef, is the edge v,v1 and 5; is the edge vju; (1 <i <n). By the 
definition of middle graph V(M(H,,)) = {v} UV(H,) U E(An) = {uy :1<i<nsUf{u:1< 
i<nbUuf{e:l<i<nbuf{e:l<i<n}U{s:1<i<n}. 


Proof Let V(An) = {v} U {v1, v2,°++ , Un} U {u1, U2,-++ , Un} and E(A,) = {e;:1<i< 
/ 





Fig.5 Middle Graph of Helm Graph M(H,,) 


Consider the following n + 1-coloring of M (H,,) as b-chromatic: For 1 <i <n, assign the 
color c; to e; and assign the color cn41 to v. For 1 <i < n, assign to vertices uj, vi, $i, €%, 
one of the allowed colors-such color exists, because deg(u;) = 1, deg(v;) = 4, deg(s;) = 3 and 
deg(e’) = 8. An easy check shows that this coloring is a b-coloring. Therefore, y(M(H,,)) > 


n+l. 


Let us assume that y(M(H,,)) is greater than n+1, i.e. p(M(HAp)) =n+2, Vn > 8, there 
must be at least n+2 vertices of degree n+1 in M(H,,), all with distinct colors, and each adjacent 
to vertices of all of the other colors. But then these must be the vertices v,{e; : 1 <i <n}, 
since these are only ones with degree at least (n — 1) + 3. This is the contradiction, b-coloring 
with n +2 colors is impossible. Thus, we have y(M(H,,)) <n+1. Hence, y(M(H,)) =n+1, 
Vn>8. 














Proposition 2.6 Ifn > 8 then b-chromatic number on total graph of Helm graph T(H,,) is 
p(T(An)) =n+1. 


Proof Consider the coloring of M (H;,) introduced on the proof of Theorem 5. An easy 
check shows that this coloring is a b-coloring of T (H,). Hence, y(T(Hn)) =n+1,V n> 8. 














122 Vernold Vivin.J., Venkatachalam M. and Mohanapriya N. 





Fig.6 Total Graph of Helm Graph T(H,,). 


Lemma 2.7 Ifn> 4 then b-chromatic number of gear graph Gy, is p(Gy) = 4. 


Proof Let V(Gr) = {vu} U {v1, v2,-++ , van} where v;’s are the vertices of cycles taken in 
cyclic order and v is adjacent with vg;-1(1 <i<7n). 


Consider the following 4-coloring (cy, c2,¢3,c4) of G, as b-chromatic: 


Assign the colors c, to v1, ¢3 to V2, C2 to U3, C1 to V4, C3 tO U5, C2 to Ve, C4 to VU and C2 to 
Von. For 7 <i < 2n—1, if any, assign to vertex v; one of the allowed colors-such color exists, 
because 2 < deg(v;) < 3. An easy check shows that this coloring is a b-coloring. Therefore, 
p(Gn) = 4. 


Let us assume that y(G,) is greater than 4, i.e. y(G,) = 5, Vn > 4, there must be at 
least 5 vertices of degree 4 in G,,, all with distinct colors, and each adjacent to vertices of all of 
the other colors. But then these must be the vertices v, {vaj_-1 : 1 < i < n}, since these are only 
ones with degree at least 3. This is the contradiction, b-coloring with 5 colors is impossible. 
Thus, we have y(G,,) < 4. Hence, y(Gn) = 4, Vn > 4. 














Lemma 2.8 [fn > 4 then b-chromatic number on line graph of Gear graph L(G,,) is p(L(Gn)) = 


n. 


Proof Let V(Gn) = {v} U {u1, v2,..., Van} and E(G,) = {e.:1<i<nbsUf{eh:1<i< 
2n —1}U {el} where e; is the edge vva;_1 (1 <i <n), ef is the edge vjvj44 (1 < i < 2n—- 1), 
and e,, is the edge van_1v1. By the definition of line graph V(L(G,,)) = E(G,) = {e5: 1 < 
i<nbUf{el:1<i< Qn}. 


On b-Chromatic Number of Some Line, Middle and Total Graph Families 123 





Fig.7 Line graph of Gear Graph L(G,,). 


Consider the following n-coloring of L (G,,) as b-chromatic: For 1 <i <n, assign the color 
c; to e;. For 1 <i < 2n, assign to vertices e/,, one of the allowed colors-such color exists, because 
deg(e’,) = 3. An easy check shows that this coloring is a b-coloring. Therefore, y(L(G,,)) > n. 

Let us assume that y(L(G,)) is greater than n, y(L(G,)) =n+1,V n> 4, there must 
be at least n + 1 vertices of degree n in L(G,,), all with distinct colors, and each adjacent to 
vertices of all of the other colors. But then these must be the vertices e;(1 <i <n), since these 
are only ones with degree at least n. This is the contradiction, b-coloring with n+ 1 colors is 
impossible. Thus, we have y(L(G,)) <n. Hence, y(L(G,)) =n, Vn > 4. 














Theorem 2.9 Ifn > 5 then b-chromatic number on middle graph of Gear graph M(G,) is 
p(M(G,)) =n+1. 


Proof Let V(Gp) = {v} U {u1, v2,+++ , van} and E(G,) = {e6: 1 <i<n}uf{eb:l<i< 
2n — 1} U {ef,} where e; is the edge vvaj_1 (1 <i < n), ef is the edge vivj4n (1 < i < 2n—-1), 
and e},, is the edge vzn_101. By the definition of middle graph V(M(G,)) = V(G,) UE(Gn) = 
{us} Uf{u:1<i< 2nbU fer: 1 <i<nhu {els 1 <i < Qn}. 





Fig.8 Middle graph of Gear Graph M(G,,). 


124 Vernold Vivin.J., Venkatachalam M. and Mohanapriya N. 


Consider the following n + 1-coloring of M (G,,) as b-chromatic: 


For 1 <i < n, assign the color c; to e; and assign the color cn+, to v. For 1 <i < 2n, 
assign to vertices v; and e,, one of the allowed colors-such color exists, because 2 < deg(v;) < 3 
and deg(e,) = 5. An easy check shows that this coloring is a b-coloring. Therefore, y(M(G,)) > 
n+l. 

Let us assume that y(M(G,,)) is greater than n+1, ie. p(M(G,)) =n+2,V n> 5, there 
must be at least n+2 vertices of degree n+1 in M(G‘,), all with distinct colors, and each adjacent 
to vertices of all of the other colors. But then these must be the vertices v,{e; : 1 <i <n}, 
since these are only ones with degree at least n+ 1. This is the contradiction, b-coloring with 
n+ 2 colors is impossible. Thus, we have y(M(G,,)) < n+ 1. Hence, p(M(G,)) = n+1, 
Vn>5. 














Proposition 2.10 Ifn > 6 then b-chromatic number on total graph of Gear graph T(G,,) is 
p(T(Gr)) =nt+1. 


Proof Consider the coloring of M (G,,) introduced on the proof of Theorem 9. An easy 
check shows that this coloring is a b-coloring of T (G,). Hence, y(T(G,)) =n+1,V n> 6. 

















Fig.9 Total graph of Gear Graph T(G,.). 


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Math. Combin. Book Ser. Vol.1(2016), 126-129 


A Note on the Strong Defining Numbers in Graphs 


Z. Tahmasbzadehbaee 


(Department of Basic Science, Technical and Vocational University, Babol branch-Alzahra, Babol, I.R. Iran) 


H.Abdollahzadeh Ahangar 


(Department of Basic Science, Babol University of Technology, Babol, I.R. Iran) 


D.A.Mojdeh 


(Department of Mathematics, University of Mazandaran, Babolsar, I.R. Iran) 
E-mail: ztahmasb@yahoo.com, ha.ahangar@nit.ac.ir, damojdehQ@umz.ac.ir 


Abstract: A defining set (of vertex coloring) of a graph G = (V, £) is a set of vertices 
S with an assignment of colors to its elements which has a unique extension to a proper 
coloring of G. A defining set S is called a strong defining set if there exists an ordering set 
{v1,V2,°++ ,Ujv|—js|} of the vertices of G—S such that in the induced list of colors in each of 
the subgraphs G—S,G—(SU{wv1}),G—(SU {v1, ve}),--- ,G@— (SU {w1, v2, +++ , vjyj—|s}—1}) 
there exists at least one vertex whose list of colors is of cardinality 1. The strong defining 
number, denoted sd(G,k), of G is the cardinality of its smallest strong defining set, where 
k > x(G). In the paper, [D.A. Mojdeh and A.P. Kazemi, Defining numbers in some of the 
Harary graphs, Appl. Math. Lett. 22 (2009), 922-926], the authors have studied the strong 
defining number in Harary graphs and posed the following problem: sd(H2m,3m+2, x) = 2m 
if m is even and sd(H2m,3m+2,xX) = 2m +1 when m is odd. In this note we prove this 


problem. 
Key Words: Defining set, strong defining set, Harary graphs. 


AMS(2010): 05C15, 05C38. 


§1. Introduction and Preliminaries 


Let G = (V,E) be a simple graph with vertex set V(G) and edge set E(G) (briefly V and E, 
respectively). The order n = n(G) of G is the number of its vertices. For every vertex v € V, 
the open neighborhood N(v) is the set {u € V | wv € E} and its closed neighborhood is the set 
N{v] = N(v)U{v}. A proper k-coloring of G is an assignment of k different colors to the vertices 
of G, such that no two adjacent vertices receive the same color. The vertex chromatic number 
of G, x(G), is the minimum number k, for which there exists a k-coloring for G. Let y(G) < 
k <|V(G)|. A set S of the vertices of G with an assignment of colors to them is called a defining 
set of vertex coloring of G, if there exists a unique extension of S$ to a proper k-coloring of G. 


A defining set with minimum cardinality is called a minimum defining set and its cardinality 


1Received April 2, 2015, Accepted February 27, 2016. 


A Note on the Strong Defining Numbers in Graphs 127 


is the defining number, denoted by d(G,k). If k = y(G), then defining number is denoted by 
d(G, x). Let G be a graph with n vertices. A defining set S, with an assignment of colors in G, 
is called a strong defining set of the vertex coloring of G with & colors if there exists an ordering 
set {U1, 02,°** ,Un—jsi} of the vertices of G—S such that in the induced list of colors in each of 
the subgraphs G—S,G—(SU{u1}), G—(SU{v1, v2}),--+ ,G—(SU{v1, v2,-++ , Un—|s|—-1}) there 
exists at least one vertex whose list of colors is of cardinality 1. The strong defining number of 
G, sd(G,k), is the cardinality of its smallest strong defining set. The strong defining number in 
graphs was introduced by Mahmoodian and Mendelsohn in [5] and has been studied by several 
authors. For more details, we refer the readers to [1-4, 6, 7]. 


For 2<k <n, the Harary graph H;,,, on n vertices is defined as follows. Place n vertices 
around a circle, equally spaced. If k is even, Hy, is formed by making each vertex adjacent 
to the nearest £ vertices in each direction around the circle. If k is odd and n is even, Hz» 
is formed by making each vertex adjacent to the nearest — vertices in each direction around 
the circle and to the diametrically opposite vertex. In both cases, Hy, is k-regular. If both k 
and n are odd, Hz, is constructed as follows. It has vertices 0, 1, --- ,2—1 and is constructed 

n=1 


from Hy—1n by adding edges joining vertex i to vertex i + “5+ for 0 < i < 45+ (see [9]). 


Mojdeh and Kazemi [8] have studied the defining and strong defining number in Harary 
graphs. In their paper, they showed that 





3m +2 
X(H2m,3m+2) = [ 2 | 
for m > 2, and posed the following conjecture. 
Conjecture A Ifn=3m+ 2, then 
2m if nis even 


8d(H2m,3m+2)X) = 
2nm+1 if nis odd. 


In this note, we prove that it is true. 


§2. Main Results 


Now we prove Conjecture A as the following Theorem. 


Theorem 2.1 [fn = 3m-+ 2, then 


2m if nis even 


8d(H2m,3m-+2)X) = 
2m+1 if nis odd. 


128 Z.Tahmasbzadehbaee, H.Abdollahzadeh Ahangar and D.A.Mojdeh 


Proof Let V(Hom,3m+2) = {£1,%2,°+* ;%3m+2}. First we show that 


2m if nis even 
sd(Ham,3m+2,X) < 
2m+1 if nis odd. 


Define the coloring function f by 


f(z) =i forl <i<[]+1, 


Or 


f(x;) =i—m—1 for 2m+3<i<[-“)]42 and 


2 
5 
f(a) =i -— 2m -—1 for [S143<i< 3m+2. 


We now consider the following cases. 
Case 1. m is even. 


Let D = {2,+++ ,@m41,2pamyyor+** ,Tpsmy4o} \ {am+2}- Clearly |D| = 2m. Consider 
the function g = f|p as an assignment of colors to D in H2m,3m+2. It is easy to see that the 
ordering 


U1, U3m+2)T3m+15°°* 1 V5] 43) C2m4+2) Up amjypys ss sUm+2 


of V(Ham,3m+2) — D satisfies the condition on definition of strong defining set and so D is a 
strong defining set of H2m,3m+2. Hence, sd(Hom,3m+2,X) < 2m in this case. 


Case 2. m is odd. 


Let D= {215 Xe, oe »Um+1, Tp 3mjto; Ae , amo} . {Zam+2}- Then |D| = 2%m+1. Let 





g = f|p be an assignment of colors to D in Ham.3m+2. Clearly f is the unique extension of g 
to a x(Ham,3m+2)-coloring. It is not hard to see that the ordering set 


{£1,€3m-+42,L3m+15°** » Tp bm) 43,0; amiyys 7° * Lm+2} 


of V(Ham,3m+2) — D satisfies the condition on definition of strong defining set and so D is a 
strong defining set of H2m,3m+2. Hence, sd(Hom,3m+2,X) < 2m+1 when m is odd. 
Now it will be shown that 


2m if nis even 
sd(Ham,3m+2,X) > : é 
2m+1 if nis odd. 


Let S be a minimum strong defining set of H = Ham.3m+2. Assume that x € V(H2m,3m+2)— 
S, then it takes the color uniquely if N(«) has at least [2] coloring vertices. Therefore, any 


[2%] vertices in S may be caused at most || + 1 of vertices in V(H2m,3m+2) — S take their 


colors uniquely, if these vertices in S are S” = {xj41, 2i42-°+* , Litem; Ty (3m) pos -Li-omti}- 
Now let m be even and S has at most 2m — 1 vertices, that is S — S$’ has 4 — 1 vertices. 


A Note on the Strong Defining Numbers in Graphs 129 


Then any vertex « € V(Ham.3m+2) — SU {Gi4m4i, Vitm+2s°°° , © 4 3m4i} has at most 
3 
> Stoyyes 


coloring vertices in N(x). This shows that the vertex x cannot take its color uniquely, a 
contradiction. Thus |.S| > 2m. 
Let m be odd and S$ has at most 2m vertices, that is S — S’ has [4] — 1 vertices. Then 


any vertex  € V(Hom,3m+2) — SU {®itm+1, Vitmt2,°°° »T4 (3m | 44} has at most 


3m 


3 


Lata x)= 2 


coloring vertices in N(x). This shows that the vertex x cannot take its color uniquely, a 











contradiction. Thus |.$| > 2m + 1 and the proof is completed. 





References 


1] H.Abdollahzadeh Ahangar and D.A.Mojdeh, On defining number of subdivided certain 
graph, Scientia Magna, 6 (2010), 110-120. 

2] D.Donovan, E.S.Mahmoodian, R.Colin and P.Street, Defining sets in combinatorics: A 
survey, in: London Mathematical Society Lecture Note Series, 307 (2003). 

3] W.A.Deuber and X.Zhu, The chromatic numbers of distance graphs, Discrete Math., 165- 
166 (1997), 195-204. 

4) E.S.Mahmoodian, Defining sets and uniqueness in graph colorings: a survey, J. Statist. 
Plann. Inference, 73 (1998), 85-89. 

5] E.S.Mahmoodian and E.Mendelsohn, On defining numbers of vertex coloring of regular 
graphs, Discrete Math., 197-198 (1999), 543-554. 

6] E.S.Mahmoodian, R.Naserasr, and M.Zaker, Defining sets of vertex coloring of graphs and 
Latin rectangles, Discrete Math., 167-168 (1997), 451-460. 

7| D.A.Mojdeh, On conjecture on the defining set of vertex graph coloring, Australas. J. 
Combin., 34 (2006), 153-160. 

8] D.A.Mojdeh and A.P.Kazemi, Defining numbers in some of the Harary graphs, Appl. Math. 
Lett., 22 (2009), 922-926. 

9] D.B. West, Introduction to Graph Theory (Second Edition), Prentice Hall, USA, 2001. 








Math. Combin. Book Ser. Vol.1(2016), 130-133 


BIOGRAPHY 


Mathematics for 
Everything with Combinatorics on Nature 
— A Report on the Promoter Dr.Linfan Mao 
of Mathematical Combinatorics 


Florentin Smarandache 

Mathematics & Science Department 
University of New Mexico 

705 Gurley Ave., Gallup, NM 87301, USA ) 


http://fs.gallup.unm.edu/FlorentinSmarandache.htm 


E-mail: fsmarandache@gmail.com 


The science’s function is realizing the natural world, developing our society in coordina- 
tion with natural laws and the mathematics provides the quantitative tool and method for 
solving problems helping with that understanding. Generally, understanding a natural thing 
by mathematical ways or means to other sciences are respectively establishing mathematical 
model on typical characters of it with analysis first, and then forecasting its behaviors, and 
finally, directing human beings for hold on its essence by that model. 

As we known, the contradiction between things is generally kept but a mathematical sys- 
tem must be homogenous without contradictions in logic. The great scientist Albert Einstein 
complained classical mathematics once that “As far as the laws of mathematics refer to reality, 
they are not certain; and as far as they are certain, they do not refer to reality.” Why did it 
happens? It is in fact result in the consistency on mathematical systems because things are full 
of contradictions in nature in the eyes of human beings, which implies also that the classical 
mathematics for things in the nature is local, can not apply for hold on the behavior of things 
in the world completely. Thus, turning a mathematical system with contradictions to a com- 
patible one and then establish an envelope mathematics matching with the nature is a proper 
way for understanding the natural reality of human beings. The mathematical combinatorics 
on Smarandache multispaces, proposed by Dr.Linfan Mao in mathematical circles nearly 10 
years is just around this notion for establishing such an envelope theory. As a matter of fact, 
such a notion is praised highly by the Eastern culture, i.e., to hold on the global behavior of 
natural things on the understanding of individuals, which is nothing else but the essence of 


combinatorics. 


1Received October 18, 2015, Accepted February 28, 2016. 


A Report on the Promoter Dr.Linfan Mao of Mathematical Combinatorics 131 


Linfan Mao was born in December 31, 1962, a worker’s family of China. After graduated 
from Wanyuan school, he was beginning to work in the first company of —it China Construc- 
tion Second Engineering Bureau at the end of December 1981 as a scaffold erector first, then 
appointed to be technician, technical adviser, director of construction management department, 
and then finally, the general engineer in construction project, respectively. But he was special 
preference for mathematics. He obtained an undergraduate diploma in applied mathematics 
and Bachelor of Science of Peking University in 1995, also postgraduate courses, such as those of 
graph theory, combinatorial mathematics, ---, etc. through self-study, and then began his ca- 
reer of doctoral study under the supervisor of Prof.Yanpei Liu of Northern Jiaotong University 
in 1999, finished his doctoral dissertation “A census of maps on surface with given underlying 
graph” and got his doctor’s degree in 2002. He began his postdoctoral research on automor- 
phism groups of surfaces with co-advisor Prof.Feng Tian in Chinese Academy of Mathematics 
and System Science from 2003 to 2005. After then, he began to apply combinatorial notion to 
mathematics and other sciences cooperating with some professors in USA. Now he has formed 
his own unique notion and method on scientific research. For explaining his combinatorial 
notion, i.e., any mathematical science can be reconstructed from or made by combinatoriza- 
tion, and then extension mathematical fields for developing mathematics, he addressed a report 
“Combinatorial speculations and the combinatorial conjecture for mathematics” in The 2nd 
Conference on Combinatorics and Graph Theory of China on his postdoctoral report “On au- 
tomorphism groups of maps, surfaces and Smarandache geometries” in 2006. It is in this report 
he pointed out that the motivation for developing mathematics in 21th century is combinatorics, 
i.e., establishing an envelope mathematical theory by combining different branches of classical 
mathematics into a union one such that the classical branch is its special or local case, or 
determining the combinatorial structure of classical mathematics and then extending classical 
mathematics under a given combinatorial structure, characterizing and finding its invariants, 
which is called the CC conjecture today. Although he only reported with 15 minutes limitation 
in this conference but his report deeply attracted audiences in combinatorics or graph theory 
because most of them only research on a question or a problem in combinatorics or graph 
theory, never thought the contribution of combinatorial notion to mathematics and the whole 
science. After the full text of his report published in journal, Prof.L.Lovasz, the chairman of 
International Mathematical Union (IMU) appraise it “an interesting paper”, and said “I agree 
that combinatorics, or rather the interface of combinatorics with classical mathematics, is a 
major theme today and in the near future” in one of his letter to Dr.Linfan Mao. This paper 
was listed also as a reference for the terminology combinatorics in Hungarian on Wikipedia, a 
free encyclopedia on the internet. After CC conjecture appeared 10 years, Dr.Linfan Mao was 
invited to make a plenary report “Mathematics after CC conjecture — combinatorial notions 
and achievements” in the International Conference on Combinatorics, Graph Theory, Topology 
and Geometry in January, 2015, surveying its roles in developing mathematics and mathemat- 
ical sciences, such as those of its contribution to algebra, topology, Euclidean geometry or 
differential geometry, non-solvable differential equations or classical mathematical systems with 
contradictions to mathematics, quantum fields and gravitational field. His report was highly 
valued by mathematicians coming from USA, France, Germany and China. They surprisingly 


132 Florentin Smarandache 


found that most results in his report are finished by himself in the past 10 years. 


Generally, the understanding on nature by human beings is originated from observation, 
particularly, characterizing behaviors of natural things by solution of differential equation es- 
tablished on those of observed data. However, the uncertainty of microscopic particles, or 
different positions of the observer standing on is resulted in different equations. For example, 
if the observer is in the interior of a natural thing, we usually obtain non-solvable differential 
equations but each of them is solvable. How can we understand this strange phenomenon? 
There is an ancient poetry which answer this thing in China, i.e., “Know not the real face of 
Lushan mountain, Just because you are inside the mountain”. Hence, all contradictions are 
artificial, not the nature of things, which only come from the boundedness or unilateral knowing 
on natural things of human beings. Any thing inherits a combinatorial structure in the nature. 
They are coherence work and development. In fact, there are no contradictions between them 
in the nature. Thus, extending a contradictory system in classical mathematics to a compatible 
one and establishing an envelope theory for understanding natural things motivate Dr.Linfan 
Mao to extend classical mathematical systems such as those of Banach space and Hilbert space 
on oriented graphs with operators, i.e., action flows with conservation on each vertex, apply 
them to get solutions of action flows with geometry on systems of algebraic equations, ordi- 
nary differential equations or partial differential equations, and construct combinatorial model 
for microscopic particles with a mathematical interpretation on the uncertainty of things. For 
letting more peoples know his combinatorial notion on contradictory mathematical systems, he 
addressed a report “Mathematics with natural reality — action flows” with philosophy on the 
National Conference on Emerging Trends in Mathematics and Mathematical Sciences of India 
as the chief guest and got highly praised by attendee in December of last year. 


After finished his postdoctoral research in 2005, Dr.Linfan Mao always used combinatorial 
notion to the nature and completed a number of research works. He has found a natural road 
from combinatorics to topology, topology to geometry, and then from geometry to theoretical 
physics and other sciences by combinatorics and published 3 graduate textbooks in mathematics 
and a number of collection of research papers on mathematical combinatorics for the guidance of 
young teachers and post-graduated students understanding the nature. He is now the president 
of the Academy of Mathematical Combinatorics & Applications (USA), also the editor-in-chief 
of International Journal of Mathematical Combinatorics (ISSN 1937-1055, founded in 2007). 


Go your own way. “Now the goal is that the horizon, Leaving the world can be only 
your back”. Dr.Linfan Mao is also the vice secretary-general of China Tendering & Bidding 
Association at the same time. He is also busy at the research on bidding purchasing policy and 
economic optimization everyday, but obtains his benefits from the research on mathematics and 
purchase both. As he wrote in the postscript “My story with multispaces” for the Proceedings 
of the First International Conference on Smarandache Multispace & Multistructure (USA) in 
2013, he said: “For multispaces, a typical example is myself. My first profession is the industrial 
and civil buildings, which enables me worked on architecture technology more than 10 years 
in a large construction enterprise of China. But my ambition is mathematical research, which 
impelled me learn mathematics as a doctoral candidate in the Northern Jiaotong University and 
then, a postdoctoral research fellow in the Chinese Academy of Sciences. It was a very strange 


A Report on the Promoter Dr.Linfan Mao of Mathematical Combinatorics 133 


for search my name on the internet. If you search my name Linfan Mao in Google, all items 
are related with my works on mathematics, including my monographs and papers published in 
English journals. But if you search my name Linfan Mao in Chinese on Baidu, a Chinese search 
engine in China, items are nearly all of my works on bids because I am simultaneously the vice 
secretary-general of China Tendering & Bidding Association. Thus, I appear 2 faces in front of 
the public: In the eyes of foreign peoples Iam a mathematician, but in the eyes of Chinese, Iam a 
scholar on theory of bidding and purchasing. So Iam a multispace myself.” He also mentioned in 
this postscript: “There is a section in my monograph Combinatorial Geometry with Applications 
to Fields published in USA with a special discussion on scientific notions appeared in TAO TEH 
KING, a well-known Chinese book, applying topological graphs as the inherited structure of 
things in the nature, and then hold on behavior of things by combinatorics on space model and 
gravitational field, gauge field appeared in differential geometry and theoretical physics. This is 
nothing else but examples of applications of mathematical combinatorics. Hence, it is not good 
for scientific research if you don’t understand Chinese philosophy because it is a system notion 
on things for Chinese, which is in fact the Smarandache multispace in an early form. There is 
an old saying, i.e., philosophy gives people wisdom and mathematics presents us precision. The 
organic combination of them comes into being the scientific notion for multi-facted nature of 
natural things on Smarandache multispaces, i.e., mathematical combinatorics. This is a kind 
of sublimation of scientific research and good for understanding the nature.” 

This is my report on Dr.Linfan Mao with his combinatorial notion. We therefore note 
that Dr.Linfan Mao is working on a way conforming to the natural law of human understand- 
ing. As he said himself: “mathematics can not be existed independent of the nature, and only 
those of mathematics providing human beings with effective methods for understanding the 
nature should be the search aim of mathematicians!” As a matter of fact, the mathematical 
combinatorics initiated by him in recent decade is such a kind of mathematics following with 
researchers, and there are journals and institutes on such mathematics. We believe that math- 
ematicians would provide us more and more effective methods for understanding the nature 
following his combinatorial notion and prompt the development of human society in harmony 
with the nature. 


134 International Journal of Mathematical Combinatorics 


Lead to something new and better. No man can sever the bonds that unite 
him to his society simply by averting his eyes. He must ever be receptive and 
sensitive to the new; and have sufficient courage and skill to novel facts and to 
deal with them. 


By Franklin Roosevelt, an American President. 


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Books 


4\Linfan Mao, Combinatorial Geometry with Applications to Field Theory, InfoQuest Press, 
2009. 
12]W.S.Massey, Algebraic topology: an introduction, Springer-Verlag, New York 1977. 


Research papers 


6|Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, International 
J.Math. Combin., Vol.3(2014), 1-34. 
9|Kavita Srivastava, On singular H-closed extensions, Proc. Amer. Math. Soc. (to appear). 





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return. 


March 2016 





Contents 


N*C* Smarandache Curve of Bertrand Curves Pair According to Frenet Frame 

By Stileyman Senyurt , Abdussamet Caliskan, Unzile Celik 

On Dual Curves of Constant Breadth According to Dual Bishop Frame in Dual 
Lorentzian Space D} By Siiha Yilmaz, Yasin Unliitiirk, Umit Ziya Saver 

On (r,m, k)-Regular Fuzzy Graphs By N.R.Santhimaheswari, C.Sekar 

Super Edge-Antimagic Labeling of Subdivided Star Trees By A.Raheem, A.Q.Baig 
Surface Family with a Common Natural Geodesic Lift By Evren Ergiin, Ergin Bayram.... 
Some Curvature Properties of LP-Sasakian Manifold with Respect to Quarter 
-Symmetric Metric Connection By Santu Dey, Arindam Bhattacharyya 

On Net-Regular Signed Graphs By Nutan G.Nayak 

On Common Fixed Point Theorems With Rational Expressions in Cone 

b-Metric Spaces’ By G.S.Saluja 

Binding Number of Some Special Classes of Trees 

By B.Chaluvaraju, H.S.Boregowda, S.Kumbinarsaiah 

On the Wiener Index of Quasi-Total Graph and Its Complement 

By B.Basavanagoud, Veena R.Desai 

Clique Partition of Transformation Graphs By Chandrakala 5.B, K.Manjula 

Probabilistic Bounds On Weak and Strong Total Domination in Graphs By M.H.Akhbari 97 
Quotient Cordial Labeling of Graphs By R.Ponraj, M.Maria Adaickalam, R.Kala 
Nonholonomic Frames for Finsler Space with (a,3)—Metrics 

By Brijesh Kumar Tripathi, K.B.Pandey, R.B. Tiwari 

On b-Chromatic Number of Some Line, Middle and Total Graph Families 

By VernoldWivine wenkatachalam Vie, Mohanaprival Nearer ee eee ee ee 116 
A Note on the Strong Defining Numbers in Graphs 

By Z.Tahmasbzadehbaee, H.Abdollahzadeh Ahangar, D.A.Mojdeh 

A Report on the Promoter Dr.Linfan Mao of Mathematical Combinatorics 


By F.Smarandache 





An International Book Series on Mathematical Combinatorics