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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 also welcome. 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Subscription A subscription can be ordered by an email directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: maolinfan@163.com Price: US$48.00 Editorial Board (4th) Editor-in-Chief Linfan MAO Shaofei Du Chinese Academy of Mathematics and System Capital Normal University, P.R.China Science, P.R.China Email: dushf@mail.cnu.edu.cn and Academy of Mathematical Combinatorics & Minodone thu Applications, USA Email: maolinfan@163.com Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Deputy Editor-in-Chief Yuanqiu Huang Hunan Normal University, P.R.China Guohua Song Email: hyqq@public.cs.hn.cn Beijing University of Civil Engineering and ; Architecture, P.R.China leer Email: songguohua@bucea.edu.cn Mansneld-UanetstiyUe Email: hiseri@mnsfld.edu Editors Xueliang Li Nankai University, P.R.China Arindam Bhattacharyya Email: lxl@nankai.edu.cn Jadavpur University, India Guodong Liu Email: bhattachar1968@yahoo.co.in Huizhou University Said Broumi Email: Igd@hzu.edu.cn Hassan IT University Mohammedia W.B.Vasantha Kandasamy Hay El Baraka Ben M’sik Casablanca Indian Institute of Technology, India B.P.7951 Morocco Email: vasantha@iitm.ac.in Junliang Cai Ion Patrascu Beijing Normal University, P-R.China Fratii Buzesti National College Email: caijunliang@bnu.edu.cn Craiova Romania Yanxun Chang Han Ren Beijing Jiaotong University, P.R.China East China Normal University, P.R.China Email: yxchang@center.njtu.edu.cn Email: hren@math.ecnu.edu.cn Jingan Cui Ovidiu-Ilie Sandru Beijing University of Civil Engineering and pojitechnica University of Bucharest Architecture, P.R.China Romania Email: cuijingan@bucea.edu.cn li International Journal of Mathematical Combinatorics Mingyao Xu Peking University, P.R.China Email: xumy@math.pku.edu.cn Guiying Yan Chinese Academy of Mathematics and System Science, P.R.China Email: yanguiying@yahoo.com Y. Zhang 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 Y.Alavi, Gary Chartrand, F.R.K.Chang, Paul Erdos, R. L.Graham and R.Ollermann, Highly irregular graphs, J. Graph Theory, 11(2) (1987), 235-249. Alison Northup, A Study of Semiregular Graphs, Bachelors thesis, Stetson University, 2002. G.S.Bloom, J.K.Kennedy and L.V.Quintas, Distance Degree Regular Graphs, The Theory and Applications of Graphs, Wiley, New York, (1981) 95-108. 4] J.A.Bondy and U.S.R.Murty, Graph Theory with Applications, MacMillan, London (1979). P.Bhattachara, Some remarks on fuzzy graphs, Pattern Recognition Lett., 6 (1987), 297- 302. K.R.Bhutani, On automorphism of fuzzy Graphs, Pattern Recognition Lett., 12 (1991), 413-420. F.Harary, Graph theory, Addition Wesley (1969). John N.Mordeson and Premchand S.Nair, Fuzzy graphs and Fuzzy Hypergraphs, Physica- Verlag, Heidelberg (2000). A.Nagoor Gani,and M.Basheer Ahamed, Order and size in fuzzy graph, Bulletin of Pure and Applied Sciences, 22E(1) (2003), 145-148. A.Nagoor Gani and S.R.Latha, On irregular fuzzy graphs, Applied Mathematical Sciences, 6, (2012), 517-523. A.Nagoor Gani and K.Radha, On regular fuzzy graphs, Journal of Physical Science, 12 (2008), 33-40. A.Rosenfeld, Fuzzy graphs, In: L.A.Zadeh,K.S.Fu, M.Shimura,Eds., Fuzzy Sets and Their Applications, Academic press (1975), 77-95. N.R. Santhi Maheswari and C.Sekar, (r,2,r(r — 1)-Regular graphs, International Journal of Mathematics and soft Computing, 2(2) (2012), 25-33. N. R.Santhi Maheswari and C.Sekar, On (2,k)-regular fuzzy graphs and totally (2, k)- regular fuzzy graphs, International Journal of Mathematics and Soft Computing , 4(2)(2014), 59-69. N.R.Santhi Maheswari and C.Sekar, On (r, 2, k)-regular fuzzy graphs, accepted in JCMCC. N.R.Santhi Maheswari and C.Sekar, On m - neighbourly irregular fuzzy graphs, Interna- tional Journal of Mathematics and Soft Computing , 5(2)(2015), 145-153. N.R.Santhi Maheswari and C.Sekar, On (m, k)-regular fuzzy graphs, International Journal 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. 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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. References 1] A.Friedmann and J.A.Schouten, Uber die Geometrie der halbsymmetrischen Uber- tra- gung, Math. Zeitschr, 21:211-223, 1924. 2] S.Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor, N.S., 29 (1975) 249-254. 3] H.A.Hayden, Subspaces of a space with torsion, Proc. London Math. Soc., 34:27-50, 1932. §.K.Hui, On ¢-pseudo symmetric Para Sasakian manifolds, Acta Universitatis Apulensis, 39 (2014), 161-178. 5] I-Mihai and R.Rosca, On Lorentzian P-Sasakian manifolds, Classical Analysis, World Sci- entific Publ., Singapore, 155-169, 1992. 6] K. Matsumoto, On Lorentzian Paracontact manifolds, Bull. Yamagata Univ. Natur. Sci., 12(2):151-156, 1989. 7| K.Matsumoto, S.lanus, I.Mihai, On P-Sasakian manifolds which admit certain tensor feilds, Publ.Math. Debrecen, 33 (1986), 61-65. 8] K.Matsumoto and I.Mihai, On a certain transformation in a Lorentzian para- Sasakian manifold, Tensor, N. S., 47:189-197, 1988. 9] R.S.Mishra, S.N.Pandey, On quarter-symmetric metric F-connection, Tensor, N.S., 34 (1980) 1-7. 10] I-Mihai and R.Rosoca, On LP-Sasakian manifolds, Classical Analysis, World Scientific Publ., (1972), 155-169. 11] S.Mukhopadhyay, A-K.Roy, B.Barua, Some properties of a quartersymmetric metric con- nection on a Riemannian manifold, Soochow J. Math., 17 (1991) 205-211. 12] G.P.Pokhariyal, R.S.Mishra, The curvature tensors and their relativisti significance, Yoko- homa Math.J., 18(1970), 105-108. 13] G.P.Pokhariyal, Study of new curvature tensor in a Sasakian manifold, Tensor, N. S., 36 (1982), 222-226. 14] S.C.Rastogi, On quarter-symmetric connection, C.R. Acad. Sci. Bulgar, 31 (1978) 811-814. 15] S.C.Rastogi, On quarter-symmetric metric connection, Tensor, N.S., 44 (1987) 133-141. 56 Santu Dey and Arindam Bhattacharyya [16] J.P.Singh, M-projective curvature tensor on LP-Sasakian manifolds, J. of Progressive Sci- ence, 3,(2012), 73-76. [17] K.Yano, On semi-symmetric metric connections, Rev. Roumaine Math. Pures Appl. 15:1579-1586, 1970. [18] K.Yano, T.Imai, Quarter-symmetric metric connections and their curvature tensors, Ten- sor, N.S., 38 (1982) 13-18. 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,.). References 1] Chen Y, Properties of spectra of graphs and line graphs, Appl. Math. J. Chinese Univ. Ser. B, 17(2002), 371-376. 2] Corteel S, Valencia-Pabon M and Vera JC, On approximating the b-chromatic number, Discrete Applied Mathematics, 146(2005), 106-110. 3] Effantin B, The b-chromatic number of power graphs of complete caterpillars, Journal of Discrete Mathematical Science & Cryptography, 8(3)(2005), 483-502. 4] Effantin B, Kheddouci H, Discussion on the (partial) Grundy and b-chromatic numbers of graphs, Utilitas Mathematica, 80(2009), 65-89. 10 11 12 13 14 15 16 17 18 19 20 On b-Chromatic Number of Some Line, Middle and Total Graph Families 125 Effantin B, Kheddouci H, The b-chromatic number of some power graphs, Discrete Math- ematics and Theoretical Computer Science, 6(2003), 45-54. Effantin B, Kheddouci H, Exact values for the b-chromatic number of a power complete k-ary tree, Journal of Discrete Mathematical Science & Cryptography, 8(1)(2005), 117-129. Faik T, About the b-continuity of graphs, Electronic Notes in Discrete Mathematics, 17(2004), 151-156. Frank Harary, Graph Theory, Narosa Publishing Home, 1969. Hamada T, Yoshimura I, Traversability and connectivity of the middle graph of a graph, Discrete Mathematics, 14(1976):247-256. Hoang C, Kouider M, On the b-dominating coloring of graphs, Discrete Applied Mathe- matics, 152(2005), 176-186. Irving RW, Manlove DF, The b-chromatic number of a graph, Discrete Appl. Math., 91(1999), 127-141. Kouider M, Maheo M, Some bounds for the b-chromatic number of a graph, Discrete Mathematics, 256(1-2)(2002), 267-277. Kratochvil J, Tuza Z, Voigt M, On the b-chromatic number of graphs, 28” International Workshop on Graph-Theoretic Concepts in Computer Science, Cesky Krumlov. Czech Republic. LNCS, 2573 (2002), 310-320. Kouider M, Zaker M, Bounds for the b-chromatic number of some families of graphs, Discrete Mathematics, 306(2006), 617-623. Kouider M , El Sahili A, About b-colouring of regular graphs, Rapport de Recherche, No.1432(2006), CNRS-Université Paris SudRI, 02/2006. Marko Jakovac, Sandi Klavzar, The b-chromatic number of cubic graphs, Graphs and Combinatorics, 26(1)(2010), 107-118. Mizuno H, Sato I, Bartholdi zeta functions of line graphs and middle graphs of graph coverings, Discrete Mathematics, 292(2005), 143-157. Sato I, Subtori of Jacobian tori associated with middle graphs of abelian regular covering graphs, Far East Journal of Applied Mathematics, 13(2003), 181-193. Venkatachalam M, Vernold VJ, The b-chromatic number of star graph families, Le Matem- atiche, 65(1)(2010), 119-125. Vernold V.J., Venkatachalam M., The b-chromatic number of corona graphs, Utilitas Math- ematica, 88(2012), 299-307. 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. Author Information Submission: Papers only in electronic form are considered for possible publication. Papers prepared in formats, viz., .tex, .dvi, .pdf, or.ps may be submitted electronically to one member of the Editorial Board for consideration both in International Journal of Mathematical Combinatorics and Mathematical Combinatorics (International Book Series). An effort is made to publish a paper duly recommended by a referee within a period of 3 months. Articles received are immediately put the referees/members of the Editorial Board for their opinion who generally pass on the same in six week’s time or less. In case of clear recommen- dation for publication, the paper is accommodated in an issue to appear next. Each submitted paper is not returned, hence we advise the authors to keep a copy of their submitted papers for further processing. Abstract: Authors are requested to provide an abstract of not more than 250 words, lat- est Mathematics Subject Classification of the American Mathematical Society, Keywords and phrases. Statements of Lemmas, Propositions and Theorems should be set in italics and ref- erences should be arranged in alphabetical order by the surname of the first author in the following style: 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). Figures: Figures should be drawn by TEXCAD in text directly, or as EPS file. In addition, all figures and tables should be numbered and the appropriate space reserved in the text, with the insertion point clearly indicated. Copyright: It is assumed that the submitted manuscript has not been published and will not be simultaneously submitted or published elsewhere. 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