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ISSN 1937 - 1055 VOLUME 1, 2015 INTERNATIONAL JOURNAL OF MATHEMATICAL COMBINATORICS EDITED BY THE MADIS OF CHINESE ACADEMY OF SCIENCES AND ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS March, 2015 Vol.1, 2015 ISSN 1937-1055 International Journal of Mathematical Combinatorics Edited By The Madis of Chinese Academy of Sciences and Academy of Mathematical Combinatorics & Applications March, 2015 Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sci- ences 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 (3nd) Editor-in-Chief Shaofei Du Linfan MAO Cepital Normal el verely, P.R.China ; : Email: dushf@mail.cnu.edu.cn Chinese Academy of Mathematics and System Science, P.R.China Baizhou He and Beijing University of Civil Engineering and Academy of Mathematical Combinatorics & Architecture, P-.R.China Applications, USA Email: hebaizhou@bucea.edu.cn Email: maolinfan@163.com Xiaodong Hu Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Deputy Editor-in-Chief Guohua Song Beijing University of Civil Engineering and Yuangiu Huang Architecture, P.R.China Hunan Normal University, P.R.China Email: songguohua@bucea.edu.cn Email: hyqq@public.cs.hn.cn H.Iseri Editors Mansfield University, USA Email: hiseriQ@mnsfld.edu S.Bhattacharya : ; ; Xueliang Li Deakin University Nankai University, P.R.China Geelong Campus at Waurn Ponds . : Email: lxl@nankai.edu.cn Australia Email: Sukanto.Bhattacharya@Deakin.edu.au Guodong Liu Huizhou University Email: Igd@hzu.edu.cn Said Broumi Hassan IT University Mohammedia Hay El Baraka Ben M’sik Casablanca W.B.Vasantha Kandasamy B.P.7951 Morocco Indian Institute of Technology, India . , Email: vasantha@iitm.ac.in Junliang Cai Beijing Normal University, P-R.China Ion Patrascu Email: caijunliang@bnu.edu.cn Fratii Buzesti National College Craiova Romania Yanxun Chang Beijing Jiaotong University, P.R.China Han Ren Email: yxchang@center.njtu.edu.cn East China Normal University, P.R.China : : Email: hren@math.ecnu.edu.cn Jingan Cui Beijing University of Civil Engineering and Ovidiu-Ilie Sandru Architecture, P.R.China Politechnica University of Bucharest Email: cuijingan@bucea.edu.cn Romania li International Journal of Mathematical Combinatorics Mingyao Xu Peking University, P.R.China Y. Zhang Email: xumy@math.pku.edu.cn Department of Computer Science Guiying Yan Georgia State University, Atlanta, USA Chinese Academy of Mathematics and System Science, P.R.China Email: yanguiying@yahoo.com Famous Words: Nothing in life is to be feared. It is only to be understood. By Marie Curie, a Polish and naturalized-French physicist and chemist. International J.Math. Combin. Vol.1(2015), 1-18 N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame Sileyman SENYURT and Abdussamet CALISKAN (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey) E-mail: senyurtsuleyman@hotmail.com Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the Mannheim curve. Besides, we illustrate example of our main results. Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet invariants. AMS(2010): 53A04 §1. Introduction 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 ([12]). Special Smarandache curves have been studied by some authors . Melih Turgut and Siitha Yilmaz studied a special case of such curves and called it Smaran- dache T Bz curves in the space E} ([12]). Ahmad T.Ali studied some special Smarandache curves in the Euclidean space. He studied Frenet-Serret invariants of a special case ([{1]). Muhammed Cetin , Yilmaz Tuncer and Kemal Karacan investigated special Smarandache curves according to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of Smarandache curves, also they found the centers of the osculating spheres and curvature spheres of Smarandache curves ({5]). Senyurt and Caligkan 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® ((2]). Nurten Bayrak, Ozcan Bektag and Salim Yiice studied some special Smarandache curves in E? [3]. Kemal Tasképrii, Murat Tosun studied special Smarandache curves according to Sabban frame on S? ({11]). In this paper, special Smarandache curve belonging to a* Mannheim partner curve such as N*C™* drawn by Frenet frame are defined and some related results are given. 1Received September 8, 2014, Accepted February 12, 2015. 2 Stileyman SENYURT and Abdussamet CALISKAN §2. Preliminaries The Euclidean 3-space E® be inner product given by (,) =a] +03 + 93 where (21,22,%3) € E®. Let a: I > E® be a unit speed curve denote by {T,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 ([6], [9]) T’ =kKN N'’=-«T+7B (2.1) Bl =-—TN. For any unit speed a : I > E3, the vector W is called Darboux vector defined by W =7(s)T(s) + «(s) + B(s). If consider the normalization of the Darboux C = wm” we have cosy = WON siny = ||| we) [WII C = sin yT(s) + cos pB(s) (2.2) where Z(W, B) = y. 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 @ and a*, respectively. If the principal normal vector N of the curve a is linearly dependent on the binormal vector B of the curve a*, then (qa) is called a Mannheim curve and (a*) a Mannheim partner curve of (a). The pair (a, a*) is said to be Mannheim pair ([7], [8]). The relations between the Frenet frames {T(s), N(s), B(s)} and {T*(s), N*(s), B*(s)} are as follows: T* = coséT — sindB N* = sin @T + cos6B (2.3) BY =N cos 9 = oor. : * dsx (2.4) sin 6 = Ar* — ds where Z(T,T*) = 0 ([8}). Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves in E® is constant. N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 3 Theorem 2.2 Let (a,a*) be a Mannheim pair curves in E?. For the curvatures and the torsions of the Mannheim curve pair (a, a*) we have, ds* = en g@—— K=7* sin ae (2.5) ds* = —7* g@— T T* cos ag and .. ad, K — dse Nr K2 + 7? (2.6) d T= (sin 0 — r cos) — S BS | Theorem 2.3 Let (a,a*) be a Mannheim pair curves in For the torsions T* of the Mannheim partner curve a* we have Theorem 2.4([{10]) Let (a,a*) be a Mannheim pair curves in E®. For the vector C* is the direction of the Mannheim partner curve a* we have g’ Ct = —— a Seca (2.7) ie Cae ie Ga) where the vector C' is the direction of the Darboux vector W of the Mannheim curve a. §3. N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame Let (a,a*) be a Mannheim pair curves in E* and {T*N*B*} be the Frenet frame of the Mannheim partner curve a* at a*(s). In this case, N*C* - Smarandache curve can be defined by 1 * * fils) = 5(N" +"). (3.1) Solving the above equation by substitution of N* and C* from (2.3) and (2.7), we obtain (cos 6|| W || + sin 04/6’? + ||W||2)T + ON + (cos 04/6”? + ||W||2 — sin || W|)B a” + || W|I? Ai(s) = (3.2) Stileyman SENYURT and Abdussamet CALISKAN The derivative of this equation with respect to s is as follows, |W ’ 9 _ 9’ «cos é shy = \|W || ‘\ || Tp, (8) (Arr) ee cst]? + = (yetthe) i |v fils OO — 2 ( Ww \ aE 4 no 4yWI) |e (eL.)'4 0/2-+4||W]|2 0 AT |W] AT||W]| /9'24\|WIl2 0’ eet wee Ww)". ee ( ts) sind] B 677+ |W? if 2 nO? +)WI2) | ( hal \e | aa ww |. WwW 2 79/2 |W? a (3.3) In order to determine the first curvature and the principal normal of the curve (1(s), we formalize v3 (ri cos @ +r sin 6)T + 73N + (—7 sin @ + 2 cos 6)B Tp, (s) = Pfr, dao Um et. ey ce « ana 2m ata (=) er 4 20? +I?) satin -2(s ML) 8 6'2+||W ||? 0’ AT||W]| AT||W || Vf 0/2 +|| W|I? 0’ where n = 2 ) (pe) Ne EA for siwie Jor iw A es ae (ena ey Ar||W'| fo? + WIP 6! forse 9 or Bt He Ww 1/9? + (IW? ie (——)’-( |) eS (iS) 6? + |W] Ar 6? + |W] o 62 + |W] eee yy |W | ee ee ea fee cd : eA oe K Pe El a la)- (2 can - Ge) N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame el ( || ) +2 6k \{( |W pee 4 a lor? + WII? XT ||W|| 62 + WII o 3 Ww 6" a | Ww chia ( 624 IW =) 9°? + ||WII2 2(=-) ( ERAT 6 x | een Sees | nn + 2 Ww 207 On \2 Ww yr V9? + IW IP Iw ( 92 4. IW =) - Ga) ae 6! | ( aT = x || Wl] > Gawd) (Gera OH Wye) e+ |W 6” + |W ||? CA Ga) Gm (Ge AT||W || 6? + ||W||2 0 / o? + |W 4 fo? + |W ia ay It 6”? sn) ~ Ga pee) 0! (2) (4 ye wi y_#_) ( We) te y (ld ——~ nS” CNS NS ~ 0? + WIP Vo? + WIP? 6? + |W IP ( OK )I( Ww | eel| MAM for eis ver +iwie 8 aa) ($) - Gain) (B)([C ial ee) ( (am) Ga): 6? + ||W||? ad )-[( ial ) av) fo? + ||WII2 6? + || WI)? e oa OK I Il y eI 6! eo tey AT||W|| or + |W | ( || W'|| )’ a” + ||W|I? ( 6! eeu 0k ) | 6 + WII? oY | Y dr Xr||W| Stileyman SENYURT and Abdussamet CALISKAN (ae g’ ———— 0! OK \? Forma, - (sa) 3 Q’ 2 OK \2 Sear, - (sa) eal ||| y ey AT gre W|I og’ eysf¢__iwh yr ve +i) iwi ao = eae I( Pie | ( wi | Ww )-25)'|( Iwi__y Jo? + |IWwIP " os yy22 AT LA fo? + | y2 ik |W ‘i (or) (eI \| | ) 0! i/o? + III? At||W I] 7 Ar gre \W|/2 el ( |W g’ i | 2 (22+ 6? + ||W||? 0 =A eae = ee 4 DS 6g’ Ok 4 OK 2 6! vereT aa - 2a) K eo Maal ace eel a " 0? + || WI fs Ky! \|W'|| 1? + (|W 7? OK \2, KF OK ce, ( ) a HG. 7) (s7) - (a) [gr 4. Maal Xr||W'| Iwi VO? +N IPD tw), ¢_ 8 al ( ) I )+ (sar) [( ) vor+imi Yo? + ie? SATII LS fo? 4 a al ( |W ae Maal )( 6! 0” + IW + (IW I2/ 4/6? + IP k WI ce, ( a” + ||W |? “[( |W | yer 6! et 6 ‘i SS | ee uly ( \w|| ) ear a YS DS + —— N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame |W] a Ok recealay, fo? + {WIP |W] alt 0? + ||W|I? 14/0 + ||W|l2 ) OK 14/0? + ||W|l2 6’ I ( g’ |W fo" + |W? Q’ ) ’ 1] 14/6!? + ||W||2 Gam) (GS) | =) ATW LN, fo? + IW ss fe? +|wie? ( | jae lel ial as “I hal ) 24+ WI) . Paywye AT US /6? + 6? + ||W|I?7? 6! 6'k \w|| 14/0? + |||? 0" | ( pres) + (serie ara a (——) +15) |(— jee Ww) (6? + WIP ATTN, fo? + || WI? @ (6? + || WI)? 0 sie Wwe yO? + IW? wi}! ( ae *)|( eae 0’ \{( aa Vf 0? + || WII? 7/ K 6'k Ok ' K OK ||W|| 6 | anand +) Gear) |( 72 = fo pt ss (oe Ok BI |W ) Sa +e. SATII! OTT 0? + WIP P+ IEP ae 6" \( oe IC hal ) ae al [9 +. ||W|l2 Ar||WI| 7 \Ar||W | Gmerye |W] fo" + ||WII? The first curvature is ——~ Ta a a Ar || W'|| : | qt + ||WI/? K ATW] V2(V ri? + 72? + 737) Ke, = 7 ( | ) ee 4 5(O7+I|W |?) 0/2-+|| Wl? go’ AT||W I >( | g! ||) 2 — ll) Vor+wies 8 Stileyman SENYURT and Abdussamet CALISKAN The principal normal vector field and the binormal vector field are respectively given by (7 cosO + fr sin 8)T + 73N + (—%4 sind + 2 cos0)B 8,5 Se (3.4) Ty + 727 + T3 _fip, &y 48 Ba (s) = ET + SN + BB. (3.5) where 622 asd Ww Ald a ie A WI ) & = re c080( et | a F2 cos 0 E 1 ENT a / , = 73 (lL) + r3( Ain) | sine [9/2 +||W ||2 T||W I by =r ( Iw ) - 0! /9!2 4.\|W ||? AT ||W Sea ( Ww )2 eee are pee "( ial ) l 3 [9°24 Wl? 0’ AT AT /9'2 4 Wl? 6’ / y —r3 |W || af P3 OK cos 0 /9'2 +||W||2 AT ||W 2 = 2 S Ww ’ Jor+ WI? : a — 9) «(0/2 4|| WII? (71? + Fo? + 737) (te) Yee | t+ (a? 4 re? + 732) 22 4M I) <t- ) a 2( WI \e AT |W] /9'2+|| WI? go’ , In order to calculate the torsion of the curve (3, we differentiate pe Ff[ooo(( ye 0? + WIP i/o? + Iwi gt te tin) fo? +lWIP fo? + we Ari vsino(( Ors ){C |W pees 6! Kk ATW LN, fo? yp Jers ive Oe ! 0”? + ||WI|I? "Kk \2 K\2 ( ial ye In a | versie, % tafe? yay SATII Oe *)-[( Thad! yr) Thal Oe meaner fo? + (WIP 6g’ (lu fo" + |W? N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame aa SH) Is -sno([( ty eo (tween) wi ae \, ( ) | (=) Zara | eC Coed de / 1/0? + ||W ||? / +o Ok IC |W || ) | ) 60 kK a a AIL fo? + IW IP ow? 47 (lu ( hal aa WI zi 6 )*-)|). Q/? i \|W||2 0 Q/? 4 |W | AT||W|| AT and thus : : my (ti cos @ + to sind + t3)T + t3N + (t2 cos 6 — ty sin@ + ts)T 1 /2 ’ where — ( Iw yp cael 0! l( Iwi__y a a, = WE fo" + ||WII? : Vo" + (|W? fo? + ||WII? wr \w|| ) desley WI g’ fo" + |W? g! fo? + |W? 7 | -|( hal peta 6! 6k ‘i owe 8 + we SATII ( OK al |W’ a 6! 6K ves Xr||W| 6? + ||W||? Q 6? + |W]? Ar||WI|7 Ar ( |W | plead |W | at 6K: ee 6k ( Ky? /o'? + WI? o /o? + |W)? Ar ||W| Ar ||WI| 7S Ar ae 2( Ok, ){C |W || yew oy 23 Ok ) Ar||W|| /o'? + |W)? ve /o'? + ||W||? Ar||W|| ( ial a, al 3 6k )( 6k ) /o'? + |W]? a’ /o'? + ||W|? Ar |W] 7 \Ar||W]| 10 Siileyman SENYURT and Abdussamet CALISKAN = OK ){C \|W’|| eas 6’ Ky IW fer simi o+|w2 47 ( WI yeaa ial al +2()|( Iwi__y Vo? + WIP ‘i 6? + WIP ver simi 7 uae La \|w| +2( maa ee “Feri fer seimie 0’ RK K i Yersime 0 = (De [ea Ee dae ee, At||W]| 7° Ar [gir Ww 0 ly? wie + WIP vere (_aw_y ve" jee ecs \/ 0? + || W|I? oe fo? + III \r ( |W yr Al |W ee 6? + WI)? f 6? + |W)? A eWay VES EIMET e SINUN — O ate ‘eae 7] | 9 LIWIP (<a) oe =.) te The torsion is then given by vp, — Heth 8 BH TEAR” Qy TB, = 208 where wy2¢__ IW I, f__ ky WP é'n_\? O = ~24(55)"(Feaes) +a ((ja ae) | a2 ~ 8 (Seq) (SO) (See + |( 4 ) wr 0” — — ——S—= — ——_—~ te Vor? + |W? Ar||W'| Vo? + WIP 0’ 0”? + ||W||? 6’ AT N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 11 |W 2 6'K |W / On \2 WII A tian + (Fae mine? | (Sav) 7 (ae ts (<3 ie (Sere) ts(57) (SF 8) 6S) - (4S) | (YY Stale) IW (2 oe + W]e, AT? NAT] 0? + WIP 0’ e+ Wz SATII a|(s wil eee) of nt |( Iw el _y ew of Vo? + ||WIP vo? + WIP e Vr + (WIP ¢ Vo? + {WIP +|( IW yer Al +(e Al ah is 1, o" )’ At L\ fo? + WIP oY Jor +wi? LS /o? +] WIP Ar 6 Ar||W'| (Ly (s,s _I_y Jerre 8 OriWy/ or er aqme! 8 Or Or WT ee = [( ol) eta) Gh) + to ( lt) ie ———— t —_— t a fe | pe +05 ((Feroep) | tat) GS) +6 Gere) ae) ae Ok tek Ok, K \2 0' ks K \3 -4( WT) (55) + Gaps) +6(sourD) Gos _({# |W Als. ||| Jer + We]? Iwi OK \? = (E(t) sl ee Ga) Iwi JOTI] oe y2¢__IWI__yIWI), ¢_O’®_)2 wy | pw y2) (_) ee 2) (see) ot Ga) E+") f(y Vem FWY ey [¢__w_y Ver FWP |wil__y’ te =| GF Five) |-Ga (ae) a ||(Geare) vere | -( OK: ) 6! ( OK. y|C WI peaeaial 0 At||WIl/ for + wy? SATII Jo? + WIP 0 -+|( ||| y ee) -( Ok JI ( ||| ) Vo? + (WIP eer ||| AT LS /0? + ||W||? oY AT||WII7 LN /6? + [WIP VO" + |W) 2 Kf OK \! |W 1/0? + |W]? 1? 0K Kar 0K -37 (sq) rt (om aa) 0" + (sq) Ge)? a (saw) ( |W yi er) g (& (= ||| — _)' Comal ||| V6? + |W? e Vo? + ae Ge 6? + WIP oY Vo? + |W —2/ 0'k )'|( ||| )’ veer) ea, Wiens ||| vl __y' wor Ar ||W|| Jo? + WIP 0’ Cre Jor + |W? 6 2 6 OK \3 Ok K\2 K\ K |W || ' Jo? + |WIP = (WT) : (seq) oe) - (x7) x5 (79 a a) Example 3.1 Let us consider the unit speed Mannheim curve and Mannheim partner curve: a(s) = —=(-—coss,—sins,s), a*(s) = —=(—2coss,—2sins,s)- W)= Fl ), a°(s) = Fel | The Frenet invariants of the partner curve, a*(s) are given as following T*(s) = —=(2sins,—2coss,1), (s) S|- 12 Siileyman SENYURT and Abdussamet CALISKAN 1 N*(s) = —s(sins,coss,—2) V5 B*(s) = (coss,sins,0) 2 2 2 2 1 C*(s) = (=sins+—<coss,—=coss + —ssins, =) 5 V5 5 VB 22 ga ee 5 In terms of definitions, we obtain special Smarandache curve, see Figure 1. i Figure 1 3; = —=((5 + 2V5) sins + 10coss, (5 — 2V5) coss + 10sins, —9V5) 5V5 References [1] Ali A.T., Special Smarandache curves in the Euclidean space, International Journal of Mathematical Combinatorics, Vol.2, 2010, 30-36. [2] Bektas O. and Yiice S., Special Smarandache curves according to Dardoux frame in Eu- clidean 3-space, Romanian Journal of Mathematics and Computer science, Vol.3, 1(2013), 48-59. [3] Bayrak N., Bektag O. and Yiice S., Special Smarandache curves in 13, International Con- ference on Applied Analysis and Algebra, 20-24 June 2012, Yildiz Techinical University, pp. 209, Istanbul. [4] Calgkan A., Senyurt S., Smarandache curves in terms of Sabban frame of spherical indi- catrix curves, XI, Geometry Symposium, 01-05 July 2013, Ordu University, Ordu. [10 (11 [12 [13 N*C*— Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 13 Cetin M., Tuncer Y. and Karacan M.K.,Smarandache curves according to bishop frame in Euclidean 3-space, arxiv:1106.3202, vl [math.DG], 2011. Hacisalihoglu H.H., Differential Geometry, Inénii University, Malatya, Mat. no.7, 1983. Liu H. and Wang F.,Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2(2008), 120-126(7). Orbay K. and Kasap E., On mannheim partner curves, International Journal of Physical Sciences, Vol. 4 (5)(2009), 261-264. Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006. Senyurt S. Natural lifts and the geodesic sprays for the spherical indicatrices of the mannheim partner curves in E?, International Journal of the Physical Sciences, vol.7, No.16, 2012, 2414-2421. Task6prii K. and Tosun M., Smarandache curves according to Sabban frame on $?, Boletim da Sociedade parananse de Mathemtica, 3 srie, Vol.32, No.1(2014), 51-59 ssn-0037-8712. Turgut M., Yilmaz S., Smarandache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, Vol.3(2008), pp.51-55. Wang, F. and Liu, H., Mannheim partner curves in 3-space, Proceedings of The Eleventh International Workshop on Diff. Geom., 2007, 25-31. International J.Math. Combin. Vol.1(2015), 14-28 Fixed Point Theorems of Two-Step Iterations for Generalized 7-Type Condition in CAT(0) 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 strong convergence theorems of modified two- step iterations for generalized Z-type condition in the setting of CAT(0) spaces. Our results extend and improve the corresponding results of [3, 6, 28] and many others from the current existing literature. Key Words: Strong convergence, modified two-step iteration scheme, fixed point, CAT(0) space. AMS(2010): 54H25, 54E40 §1. Introduction A metric space X is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having non-positive sectional curvature is a CAT(0) space. Fixed point theory in a CAT(0) space was first studied by Kirk (see [19, 20]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since, then the fixed point theory for single-valued and multi-valued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2], [9], [11]-[13], [17]-[18], [21]-[22], [24]- [26] and references therein). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT(k) space with k < 0 since any CAT(k) space is a CAT(m) space for every m > k (see [7). Let (X,d) be a metric space. A geodesic path joining x € X to y € X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,/] C R to X such that c(0) = a, c(l) = y and d(c(t),c(t’)) = |t — t’| for all t,t’ € [0,/]. In particular, c is an isometry, and d(x,y) =1. The image a of c is called a geodesic (or metric) segment joining x and y. We say X is (i) a geodesic space if any two points of X are joined by a geodesic and (ii) a uniquely geodesic if there is exactly one geodesic joining x and y for each x,y € X, which we will denoted by [x,y], called the segment joining x to y. A geodesic triangle A(a1, 22,23) in a geodesic metric space (X,d) consists of three points 1Received July 16, 2014, Accepted February 16, 2015. Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 15 in X (the vertices of A) and a geodesic segment between each pair of vertices (the edges of A). A comparison triangle for geodesic triangle A(x1, 72,23) in (X,d) is a triangle A(x, v2, 73) := A (%q,%2,%3) in R* such that dp2(%j,7j) = d(x;,x;) for i,j € {1,2,3}. Such a triangle always exists (see [7]). 1.1 CAT(0) Space A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following CAT (0) comparison axiom. Let A be a geodesic triangle in X, and let A C R? be a comparison triangle for A. Then A is said to satisfy the CAT (0) inequality if for all 2, y € A and all comparison points 7,9 € A, Complete CAT(0) spaces are often called Hadamard spaces (see [16]). If x,y1, y2 are points of a CAT(0) space and yo is the mid point of the segment [yi, y2] which we will denote by (y1 @ y2)/2, then the CAT (0) inequality implies ® 1 1 1 P (0, A) <5 Pam) +5 Pew) — 5 Hvry). (1.2) The inequality (1.2) is the (CN) inequality of Bruhat and Tits [8]. The above inequality was extended in [12] as @(z,ax@(l—a)y) < ad?(z,x) +(1—a)d?(z,y) —a(1 — a)d? (x,y) (1.3) for any a € [0,1] and z,y,z€ X. Let us recall that a geodesic metric space is a CAT (0) space if and only if it satisfies the (CN) inequality (see [7, page 163]). Moreover, if X is a CAT (0) metric space and x,y € X, then for any a € [0,1], there exists a unique point ax © (1 — a)y € [x,y] such that d(z,axz @ (1—a)y) < ad(z,x) + (1— a)d(z,y), (1.4) for any z € X and [x,y] = {ax 6 (1—a)y: a€ (0, 1]}. A subset C of a CAT (0) space X is convex if for any x,y € C, we have [x,y] C C. We recall the following definitions in a metric space (X,d). A mapping T: X — X is called an a-contraction if d(Tx,Ty) <ad(x,y) for allx, y € X, (1.5) where a € (0,1). The mapping T is called Kannan mapping [15] if there exists b € (0,4) such that d(Tx,Ty) < b[d(a,Tx) + d(y,Ty)| (1.6) for all a,y € X. 16 G.S.Saluja The mapping T is called Chatterjea mapping [10] if there exists c € (0,4) such that d(Tx,Ty) < c[d(x, Ty) + dy, Tx)] (1.7) for all a,y € X. In 1972, Zamfirescu [29] proved the following important result. Theorem Z Let (X,d) be a complete metric space and T: X — X a mapping for which there exists the real number a, b and c satisfying a € (0,1), b, c € (0,4) such that for any pair zx, y€ X, at least one of the following conditions holds: (a1) d(Tx,Ty) < ad(x,y); (22) d(L'x,Ty) < b|d(x, Tx) + dy, Ty)}; (23) d(L'x,Ty) < cld(x, Ty) + dy, Tx)}. Then T has a unique fixed point p and the Picard iteration {x,}°° defined by En+1 = Lary, n= OFT 2 ee: converges to p for any arbitrary but fixed xp € X. An operator T' which satisfies at least one of the contractive conditions (z1), (z2) and (z3) is called a Zamfirescu operator or a Z-operator. In 2004, Berinde [5] proved the strong convergence of Ishikawa iterative process defined by: for xo € C, the sequence {z,,}°° given by Ungi = (L—an)an + OnT Yn, Yn = (1 _ Bn )&n + PnT%y, n=O, (1.8) to approximate fixed points of Zamfirescu operator in an arbitrary Banach space E. While proving the theorem, he made use of the condition, Px —Tyl| <6 |jx— yl] + 25 lw — To (1.9) which holds for any z, y € E where0< 6 < 1. In 1953, W.R. Mann defined the Mann iteration [23] as Un+1 = (1 = An)Un + anT Un, (1.10) where {a,,} is a sequence of positive numbers in [0,1]. In 1974, S.Ishikawa defined the Ishikawa iteration [14] as Sn41 = (1— dn)$n + OnTtn, th = (1— bn) Sn + bnT Sn, (1.11) where {a,,} and {b,} are sequences of positive numbers in [0,1]. Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 17 In 2008, S.Thianwan defined the new two step iteration [27] as Unt = (1—an)Wn + OnT wn, Wn = (1 = bn) + OnT Vn, (1.12) where {a,,} and {b,} are sequences of positive numbers in [0,1]. Recently, Agarwal et al. [1] introduced the S-iteration process defined as Engi = (1L—an)T Ln + OnT Yn, Yn = (1 — bn) &n + bnT an, (1.13) where {a,,} and {b,,} are sequences of positive numbers in (0,1). In this paper, inspired and motivated [5, 29], we employ a condition introduced in [6] which is more general than condition (1.9) and establish fixed point theorems of S- iteration scheme in the framework of CAT(0) spaces. The condition is defined as follows: Let C be a nonempty, closed, convex subset of a CAT(0) space X and T: C — C a self map of C. There exists a constant L > 0 such that for all x, y € C’, we have d(Tx,Ty) < e& UT) [6 d(x,y) + 26d(x,Tx)], (1.14) where 0 < 6 < 1 and e® denotes the exponential function of « € C. Throughout this paper, we call this condition as generalized Z-type condition. Remark 1.1 If ZL =0, in the above condition, we obtain d(Tx,Ty) < dd(a,y) + 26 d(a,T x), which is the Zamfirescu condition used by Berinde [5] where b Cc 6 max {a, [3° —}, 0<6<1, while constants a, b and c are as defined in Theorem Z. Example 1.2 Let X be the real line with the usual norm |].|| and suppose C' = [0,1]. Define T:C > C by Tx = xit for all x,y € C. Obviously T is self-mapping with a unique fixed point 1. Now we check that condition (1.14) is true. If v,y € [0,1], then ||[Ta—Ty]| < ef llz@—Tall§ || — yl] + 26 ||a — Ta||] where 0 < 6 < 1. In fact |T2—Tyl| = t—y 2 and -—1 eblleTel 5 a — yll + 26 lle — Ta] = e* lel [51a — yl +5 Ie — 1]. 18 G.S.Saluja Clearly, if we chose x = 0 and y = 1, then contractive condition (??) is satisfied since 1 Tx —Ty|| = == [Tx - Ty 7 t—y 2 and for L > 0, we chose L = 0, then eb lle-Tel | § 1a — yl] + 26 le — Ta] ] = e* lll [5 Ja — yl +5 a — 111] = e9(1/2)(26) = 25, where 0<6 <1. Therefore Px — Tyl| < oP !#- 72H] |] — yl| + 26 x — Tal]. Hence T is a self mapping with unique fixed point satisfying the contractive condition (1.14). Example 1.3 Let X be the real line with the usual norm |].|| and suppose K = {0,1, 2,3}. Define T: K — K by Tzr=2, if «=0 = 3, otherwise. Let us take x = 0, y= 1 and L =0. Then from condition (1.14), we have 1 IA e°(?)15(1) + 26(2)] < 1(56) = 56 which implies 6 > ¢. Now if we take 0 < 6 < 1, then condition (1.14) is satisfied and 3 is of course a unique fixed point of T. 1.2 Modified Two-Step Iteration Schemes in CAT(0) Space Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let T: C > C be a contractive operator. Then for a given x1 = x € C, compute the sequence {z,,} by the iterative scheme as follows: Ung = (1L—an)T Ln B OnT Yn, Yn = (1 = bn) &n O bn» T In, (1.15) where {a,,} and {b,} are sequences of positive numbers in (0,1). Iteration scheme (1.15) is called modified S-iteration scheme in CAT(0) space. Vnti1 = (1—an)Wn 8 OnT wn, Wn = (1 = bn)in 8 OnT Vn, (1.16) where {a,,} and {b,,} are sequences of positive numbers in [0,1]. Iteration scheme (1.16) is called Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 19 modified $.Thianwan iteration scheme in CAT(0) space. Sn41 = (1— dn)Sn ® AnTtn, th = (1— bn) Sn ® bnT Sn, (1.17) where {a,,} and {b,,} are sequences of positive numbers in [0,1]. Iteration scheme (1.17) is called modified Ishikawa iteration scheme in CAT(0) space. We need the following useful lemmas to prove our main results in this paper. Lemma 1.4([24]) Let X be a CAT(0) space. (i) Fora, y€X andt € [0,1], there exists a unique point z € [x, y] such that d(x, 2) =td(a, y) anddly, 2) = (1—#)d{a, y). (4) We use the notation (1—t)x @ ty for the unique point z satisfying (A). (ii) Fora, ye X andt € [0,1], we have d((1 —t)a @ ty, z) < (1—t)d(a, z) + td(y, z). Lemma 1.5([4]) Let {pn }°o, {an}%o, {rn} eo be sequences of nonnegative numbers satisfying the following condition: Pnti <(1—Sn)Pn tanttn, Vn>0, where tate COO AY df Sag ga = 00, lit, 355Gn = Og) and >) arn = tO, hen §2. Strong Convergence Theorems in CAT(0) Space In this section, we establish some strong convergence theorems of modified two-step iterations to converge to a fixed point of generalized Z-type condition in the framework of CAT(0) spaces. Theorem 2.1 Let C be a nonempty closed convex subset of a complete CAT(0) space X and let T: C > C be a self mapping satisfying generalized Z-type condition given by (1.14) with F(T) #0. For any xo € C, let {an}°29 be the sequence defined by (1.15). If 07-4 an = 00 and pete Anbn = 00, then {rn }°2_ converges strongly to the unique fixed point of T. Proof From the assumption F(T) 4 @, it follows that T has a fixed point in C, say uw. Since T satisfies generalized Z-type condition given by (1.14), then from (1.14), taking x = u 20 G.S.Saluja and y = tn, we have Wien) eter) (6 d(u, @,) + 26 d(u, Tu) = eb dun) (6 d(u, tn) + 26 d(u, u)) eb (0) (6 d(u,a,) +26 (0) , which implies that d(Tan,u) < dd(an,u). (2.1) Similarly by taking « = u and y = yp in (1.14), we have Now using (1.15), (2.2) and Lemma 1.4(ii), we have d(yn,u) = d((1— dp)an O bnT Xn, U) < (1—b,)d(ap,u) + by d(T an, u) < (1—b,)d(an,u) + 6,6 d(an, u) = (L—bn 4 dn d)d(an,u). (2.3) Now using (1.15), (2.1), (2.3) and Lemma 1.4(ii), we have Utnpi,u) = d((1—an)Tan ® anT Yn, U) < (1—an)d(Tan,u) + and(Tyn, u) S (1—an)6d(an,u) + and d(yn, u) ra I (1 — d)an|d(an,u) + and[1 — (1 — 5) bp |d(an, u) 1— (1 d)an + an d(1 — (1 — 6)b,)]d(tp, u) 1— {(1 — d)an + 6(1 — db) an bp }d(ap, u) = (1 — pn) d(an, u) (2.4) ( ( (1 — an + Gy 0)d(an, U) + an O(1 — by + dy 6)d(Hp, U) ee [ [ where fin = (1 — d)an + 6(1 — d)anbn. Since 0 < 6 < 1; an, bn € (0,1); 2p an = 00 and rg anbn = ©, it follows that 377° 4 fin = 00. Setting py, = d(atn,U), Sn = fn and by applying Lemma 1.5, it follows that limp... d(an,u) = 0. Thus {x,}°25 converges strongly to a fixed point of T. To show uniqueness of the fixed point u, assume that ui, ue € F(T) and wu, 4 uz. Applying generalized Z-type condition given by (1.14) and using the fact that 0 < 6 < 1, we obtain d(ui,u2) = d(Pur, Tus) ef TONS d(uy, u2) + 26 d(ur, Tu:)} IA es ef dura) |S d(uy, us) + 25d(ur, 1) Fixed Point Theorems of Two-Step Iterations for Generalized Z-Type Condition in CAT(0) Spaces 21 S ec ()f 5 d(ur, uz) + 25(0)} = dd(ur, U2) < d(ui, ug), which is a contradiction. Therefore u; = wa. Thus {x,,}°2,) converges strongly to the unique fixed point of T. Theorem 2.2 Let C be a nonempty closed convex subset of a complete CAT(0) space X and let T: C > C be a self mapping satisfying generalized Z-type condition given by (1.14) with F(L) £0. For any xo € C, let {xn} be the sequence defined by (1.16). If Sy 9 an = ~, then {%n}°_ converges strongly to the unique fixed point of T. Proof The proof of Theorem 2.2 is similar to that of Theorem 2.1. Theorem 2.3 Let C be a nonempty closed convex subset of a complete CAT(0) space X and let T: C > C be a self mapping satisfying generalized Z-type condition given by (1.14) with F(T) #0. For any xo € C, let {xn} 9 be the sequence defined by (1.17). If Py an = and yy 9 anbn = 00, then {xn} converges strongly to the unique fixed point of T. Proof The proof of Theorem 2.3 is also similar to that of Theorem 2.1. If we take L = 0 in condition (1.14), then we obtain the following result as corollary which extends the corresponding result of Berinde [5] to the case of modified S-iteration scheme and from arbitrary Banach space to the setting of CAT(0) spaces. Corollary 2.4 Let C be a nonempty closed convex subset of a complete CAT(0) space X and let T: C > C a Zamfirescu operator. For any xo € C, let {an}%2 9 be the sequence defined by (1.15). If Sy an = 0 and YY 9 anbn = 00, then {xn} converges strongly to the unique fixed point of T. Remark 2.5 Our results extend and improve upon, among others, the corresponding results proved by Berinde [3], Yildirim et al. [28] and Bosede [6] to the case of generalized Z-type condition, modified S-iteration scheme and from Banach space or normed linear space to the setting of CAT(0) spaces. §3. Conclusion The generalized Z-type condition is more general than Zamfirescu operators. Thus the results obtained in this paper are improvement and generalization of several known results in the existing literature (see, e.g., [3, 6, 28] and some others). References [1] R.P.Agarwal, Donal O’Regan and D.R.Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Nonlinear Convex Anal. 8(1)(2007), 61-79. 22 10 11 12 13 14 15 16 17 18 19 [20 (21 G.S.Saluja A.Abkar and M.Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal.: TMA, 74(5)(2011), 1835-1840. 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Vol.1(2015), 24-34 Antidegree Equitable Sets in a Graph Chandrashekar Adiga and K. N. Subba Krishna (Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India) E-mail: c_adiga@hotmail.com; sbbkrishnaQ@gmail.com Abstract: Let G = (V, E) beagraph. A subset S$ of V is called a Smarandachely antidegree equitable k-set for any integer k, 0 < k < A(G), if |deg(u) — deg(v)| # k, for all u,v € S. A Smarandachely antidegree equitable 1-set is usually called an antidegree equitable set. The antidegree equitable number AD.(G), the lower antidegree equitable number ad.(G), the independent antidegree equitablenumber AD;-(G) and lower independent antidegree equitable number adie(G) are defined as follows: AD.(G) = max{|S|: S is a maximal antidegree equitable set in G}, ade(G) = min{|S|: S is a maximal antidegree equitable set in G}, ADie(G) = max{|S|: S is a maximal independent and antidegree equitable set in G}, adie(G) = min{|S|: S is a maximal independent and antidegree equitable set in G}. In this paper, we study these four parameters on Smarandachely antidegree equitable 1-sets. Key Words: Smarandachely antidegree equitable k-set, antidegree equitable set, antide- gree equitable number, lower antidegree equitable number, independent antidegree equitable number, lower independent antidegree equitable number. AMS(2010): 05C69 §1. Introduction By a graph G = (V,£) we mean a finite, undirected graph with neither loops nor multiple edges. The number of vertices in a graph G is called the order of G and number of edges in G is called the size of G. For standard definitions and terminologies on graphs we refer to the books [2] and [3]. In this paper we introduce four graph theoretic parameters which just depend on the basic concept of vertex degrees. We need the following definitions and theorems, which can be found in [2] or [3]. Definition 1.1 A graph G is isomorphic to a graph G2, if there exists a bijection o from V(Gi) to V(G2) such that uv € E(G)) if, and only if, d(u)d(v) € E(G2). If G, is isomorphic to G2, we write G; = G2 or sometimes G, = Go. 1Received June 16, 2014, Accepted February 18, 2015. Antidegree Equitable Sets in a Graph 25 Definition 1.2 The degree of a vertex v in a graph G is the number of edges of G incident with v and is denoted by deg(v) or degg(v). The minimum and maximum degrees of G are denoted by 6(G) and A(G) respectively. Theorem 1.3 In any graph G, the number of odd vertices is even. Theorem 1.4 The sum of the degrees of vertices of a graph G is twice the number of edges. Definition 1.5 The corona of two graphs G; and Gz is defined to be the graph G = G, 0 G2 formed from one copy of Gy and |V(G4)| copies of Gz where the i” vertex of Gy is adjacent to every vertex in the i*” copy of Go. Theorem 1.6 Let G be a simple graph 1.e, a undirected graph without loops and multiple edges, with n > 2. Then G has atleast two vertices of the same degree. Definition 1.7 Any connected graph G having a unique cycle is called a unicyclic graph. Definition 1.8 A graph is called a caterpillar if the deletion of all its pendent vertices produces a path graph. Definition 1.9 A subset S of the vertex set V in a graph G is said to be independent if no two vertices in S are adjacent in G. The maximum number of vertices in an independent set of G is called the independence number and is denoted by ((G). Theorem 1.10 Let G be a graph and S CV. S is an independent set of G if, and only if, V —S is a covering of G. Definition 1.11 A clique of a graph is a maximal complete subgraph. Definition 1.12 A clique is said to be maximal if no super set of it is a clique. Definition 1.13 The vertex degrees of a graph G arranged in non-increasing order is called degree sequence of the graph G. Definition 1.14 For any graph G, the set D(G) of all distinct degrees of the vertices of G is called the degree set of G. Definition 1.15 A sequence of non-negative integers is said to be graphical if it is the degree sequence of some simple graph. Theorem 1.16([1]) Let G be any graph. The number of edges in G% the degree equitable graph A-1 A > Ge = > ) ime \ 2 i=5+41 7 where, S; ={v|v € V, deg(v) =i or i+1} and S;' = {v|v € V, deg(v) = i}. of G, is given by 26 Chandrashekar Adiga and K. N. Subba Krishna Theorem 1.17 The maximum number of edges in G with radius r > 3 is given by n? — 4nr + 5n + 4r? — 6r Definition 1.18 A vertex cover in a graph G is such a set of vertices that covers all edges of G. The minimum number of vertices in a vertex cover of G is the vertex covering number a(G) of G. Recently A. Anitha, S. Arumugam and E. Sampathkumar [1] have introduced degree eq- uitable sets in a graph and studied them. “The characterization of degree equitable graphs” is still an open problem. In this paper we give some necessary conditions for a graph to be degree equitable. For this purpose, we introduce another concept “Antidegree equitable sets” in a graph and we study them. §2. Antidegree Equitable Sets Definition 2.1 Let G = (V,E) be a graph. A non-empty subset S of V is called an antidegree equitable set if |\deg(u) — deg(v)| 4.1 for allu,v eS. Definition 2.2 An antidegree equitable set is called a maximal antidegree equitable set if for every v€ V —S, there exists at least one element u € S such that |\deg(u) — deg(v)| = 1. Definition 2.3 The antidegree equitable number AD.(G) of a graph G is defined as AD.(G) = max{|S|:S is a maximal antidegree equitable set}. Definition 2.4 The lower antidegree equitable number ad-(G) of a graph G is defined as ad.(G) = min{|S|: S is a maximal antidegree equitable set}. A few AD.(G) and ad.(G) of some graphs are listed in the following: (i) For the complete bipartite graph Kin.n, we have + if |m—n| £1, AD. (Kina) = mtn if |m | A max{m,n} if |m—n| =1 and m+n if |m—n| £1, ade(Km mn) = | |# mintm,n} if |m—n| =1. (ii) For the wheel W,, on n-vertices, we have n ifn #5, AD.(W,,) = 4 ifn=5 Antidegree Equitable Sets in a Graph 27 and n ifn #5, ad-(W,,) = 1 ifn=5. (iti) For the complete graph K,, we have AD.(Kn) = ade(Kyn) =n-— 1. Now we study some important basic properties of antidegree equitable sets and independent antidegree equitable sets in a graph. Theorem 2.5 Let G be a simple graph on n-vertices. Then i) 1<ad-(G) < AD.(G) <n; it) AD.(G) =1 if, and only if, G= ky; iit) ade(G) = ad.(G), AD.(G) = AD.(G). iv) ade(G) = 1 if, and only if, there exists a verter u € V(G) such that |deg(u)—deg(v)| = 1 for allvu € V — {u}; (v) If G is a non-trivial connected graph and ad.(G) = 1, then AD.(G) =n—-1 andn must be odd. —~ ~~ a Proof (i) follows from the definition. (ii) Suppose AD. (G) = 1 and G 4 Ky. Then G is a non-trivial graph and from Theorem 1.6 there exists at least two vertices of same degree and they form an antidegree equitable set in G. So AD.(G) > 2 which is a contradiction. The converse is obvious. (iii) Since degg(u) = (n — 1) — degg(u), it follows that an antidegree equitable set in G is also an antidegree equitable set in G. (iv) If ad.(G) = 1 and there is no such vertex u in G, then {wu} is not a maximal antidegree equitable set for any u € V(G) and hence ad.(G) > 2 which is a contradiction. The converse is obvious. (v) Suppose G is a non-trivial connected graph with ad.(G) = 1. Then there exists a vertex u € V such that |deg(u) — deg(v)| = 1, V v € V — {u}. Clearly, |deg(v) — deg(w)| = 0 or 2, Vu, we V—{u}. Hence, AD.(G) = |V — {u}| = n — 1. It follows from Theorem 1.4 that (n — 1) is even and thus n is odd. Theorem 2.6 Let G be a non-trivial connected graph on n-vertices. Then 2 < AD.(G) <n and AD.(G) = 2 if, and only if, G = Ky or Py or P3 or L(H) or L?(H) where H is the caterpillar Ts with spine P = (v1 v2). Proof By Theorem 2.5, for a non-trivial connected graph G on n-vertices, we have 2 < AD,.(G) <n. Suppose AD,.(G) = 2. Then for each antidegree equitable set S in G, we have |S| < 2. Let D(G) = {di, do,...,dx}, where dy < dz < dg < --- < dx. As there are at least two vertices with same degree, we have k < n— 1. Since AD.(G) = 2, more than two vertices cannot have the same degree. Let d; € D(G) be such that exactly two vertices of G have degree d;. Since the cardinality of each antidegree equitable set S cannot exceed two, it follows that 28 Chandrashekar Adiga and K. N. Subba Krishna -++ ,d;—3,d;—2,d;+2,d;+3,d;+4,--- do not belong to D(G). Thus D(G) e {d;—1, di, d;+1}. Case 1. If d; —1,d; +1 do not belong to D(G) then D(G) = {d;} and the degree sequence {d;,d;} is clearly graphical. Thus n = 2 and d; = 1 which implies G = Ko. Case 2. If d; —1,d; +1 € D(G), then the degree sequence {d; — 1, d;, d;,d; + 1} is graphical. Thus n = 4 and d; = 2 which implies G & D(H), where H is the caterpillar T; with spine P= (v1 v2). Case 3. Ifd;—1 € D(G) and d;+1 does not belong to D(G), then d;—1 may or may not repeat twice in degree sequence. Thus degree sequence is given by {d;—1, d;, d;} or {d;—1, d;—1, d;, di}. The first sequence is not graphical but the second sequence is graphical. Thus n = 4 and d; = 2 which implies G = P,. Case 4. If d; — 1 does not belong to D(G) and d; +1 € D(G), then the degree sequence is given by {d;, d;,d; + 1} or {d;, d;,d; + 1,d; +1}. Both sequences are graphical. In the first case n = 3, d; = 1 which implies G © P, and in the second case n = 4, d; = 1 or 2 which implies G & Ps or G& L?(H) respectively. The converse is obvious. Theorem 2.7 Ifa and b are positive integers with a < b, then there exists a connected simple graph G with ad.(G) =a and AD.(G) = b except when a= 1 andb=2m+1,meEN. Proof If a = b then for any regular graph of order a, we have ad.(G) = AD,.(G) = a. If b = a+1, then for the complete bipartite graph G = kaa+1 we have ad.(G) = a and AD.(G) =a+1=b). Ifb >a+2,a> 2, and b > 4, then for the graph G consisting of the wheel W,_1 and the path P, = (v1v2v3...Uqa) with an edge joining a pendant vertex of P, to the center of the wheel Wy_1, we have ad.(G) = a, AD-(G) = b. Ifa =1andb=2m,meN, then the graph consisting of two cycles Cy, and C41 along with edges joining i'” vertex of Cm to i” vertex of C41, we have ad.(G) = 1 =a and AD,(G) = 2m=b. Figure 1 Antidegree Equitable Sets in a Graph 29 For a = 2 and b = 4 we consider graph G in Figure 1, for which ad.(G) = 2 and AD.(G) = 4. Also, it follows from Theorem 2.5 that there is no graph G with ad.(G) = 1 and AD.(G) = 2m +1. Theorem 2.8 Let G be a non-trivial connected graph on n vertices and let S* be a subset of V A— such that |deg(u) — deg(v)| > 2 for allu,v € S*. Then 1 < |S*| < AS] +1 and also, if S* is a maximal subset of V such that |deg(u)—deg(v)| > 2 for allu, v € S*, then S = U Daeg) vEs* is a maximal antidegree equitable set in G, where Saeg(v) = {u € V : deg(u) = deg(v)}. Proof For any two vertices u,v € S*, d(u) and d(v) cannot be two successive members of A= {6,6+1,6+4+2,...,6 +k =A} and D(G) c A. Hence \s*|< [Pi] : aS _ AS] ea Ifa,be S =U, .5+ Sadeg(v), then it is clear that either |deg(a)—deg(b)| = 0 or |deg(a)—deg(b)| = 2 and hence S' is an antidegree equitable set. Suppose u € V — S. Then deg(u) 4 deg(v) for any v € S*. So, u do not belong to S* and hence |deg(u) — deg(v)| = 1 for all v € S. This implies that S$ is a maximal antidegree equitable set. Theorem 2.9 Given a positive integer k, there exists graphs G, and G2 such that ad-(G1) — ad.(G, — e) =k and ad. (G2 — e) — ad. (G2) = k. Proof Let Gi = Kpy2. Then ad.(Gi) = k +2 and ad.(G; — e) = 2, where e € E(G;). Hence ad.(G1) — ad.(G1 — e) = k. Let G2 be the graph obtained from C,41 by attaching one leaf e at (k + 1)!” vertex of Cy41. Then ad-(Gz — e) — ade(Gz) = k. Theorem 2.10 Given two positive integers n and k withk <n. Then there exists a graph G of order n with ade(G) =k. Proof If k < 4, then we take G' to be the graph obtained from the path P,, = (v1v2v3..- vr) and the complete graph K,_, by joining v; and a vertex of K,_, by an edge. Clearly, ad.(G) = k. If k > $, then we take G to be the graph obtained from the cycle Cy by attaching exactly one leaf at (n — k) vertices of Cy. Clearly, ad.(G) = k. §3. Independent Antidegree Equitable Sets In this section, we introduce the concepts of independent antidegree equitable number and lower independent antidegree equitable number and establish important results on these parameters. Definition 3.1 The independent antidegree equitable number ADje(G) = maz{|S| : 8 Cc V,S is a maximal independent and antidegree equitable set in G}. Definition 3.2 The lower independent antidegree equitable number adie(G) = min{|S| : 30 Chandrashekar Adiga and K. N. Subba Krishna S is a maximal independent and antidegree equitable set in G}. A few ADj- and adje of graphs are listed in the following. (i) For the star graph Ky,, we have, ADie(Ki,) =n and adje(Ki.n) = 1. (it) For the complete bipartite graph Km» we have ADj;e(Kmn) = max{m,n} and adie(Kmn) = min{m, n}. (iti) For any regular graph G we have, ADje(G) = adie(G) = (o(G). The following theorem shows that on removal of an edge in G, AD;-(G) can decrease by at most one and increase by at most 2. Theorem 3.3 Let G be a connected graph, e = uv € E(G). Then ADie(G) —1 < ADie(G cary e) < AD ie(G) + 2. Proof Let S be an independent antidegree equitable set in G with |S| = AD;.(G). After removing an edge e = uv from the graph G, we shall give an upper and a lower bound for ADie(G = e). Case 1. If u,v does not belong to S, then S is a maximal independent antidegree equitable set in G—e as well as in G. Hence, AD;.(G — e) = AD; (G). Case 2. If u€ S and v does not belong to S, then S — {u} is an independent antidegree equitable set in G—e. Hence, AD;.(G — e) > |S — {u}| = AD;-(G) — 1. Thus, AD;.(G) — 1 < ADie(G = e). Now, Let S be an independent antidegree equitable set in G — e with |.S| = AD;-(G — e). Case 3. If u,v € S, then S — {u,v} is an independent antidegree equitable set in G. Hence, by definition AD;.(G) > |S — {u, v}| = ADie(G — e) — 2. Case 4. If u€ S$ and v does not belong to S, then S — {u} is an independent antidegree equitable set in G. Hence, by definition AD;.(G) > |S — {u}| = ADie(G — e) — 1. Case 5. If u, v do not belong to S, then S is an independent antidegree equitable set in G. Hence, by definition AD;.(G) > |S| = AD;-(G—e). It follows that AD;.(G) > ADie(G—e) —2. Hence, ADie(G) — 1 < ADie(G — e) < ADie(G) 4+ 2. Theorem 3.4 Let G be a connected graph. ADie(G) = 1 if, and only if, G= Ky, or for any two non-adjacent vertices u,v € V, |deg(u) — deg(v)| = 1. Proof Suppose ADj.(G) = 1. Case 1. If G=K,, then there is nothing to prove. Case 2. Let G #4 Ky, and u, v be any two non-adjacent vertices in G. Since AD;.(G) = 1, {u,v} is not an antidegree equitable set and hence |deg(u) — deg(v)| = 1. The converse is Antidegree Equitable Sets in a Graph 31 obvious. Theorem 3.5 Let G be a connected graph. adie(G) = 1 if, and only if, either A = n—1 or for any two non-adjacent vertices u,v € V, |deg(u) — deg(v)| = 1. Proof Suppose ad;-(G) = 1, then for any two non-adjacent vertices u and v, {u,v} is not an antidegree equitable set. Case 1. If A=n-—1, then there is nothing to prove. Case 2. Let A < n—1, and u, v be any two non-adjacent vertices in G. Then {u,v} is not an antidegree equitable set and hence, |deg(u) — deg(v)| = 1. The converse is obvious. Remark 3.6 Theorems 3.4 and 3.5 are equivalent. §4. Degree Equitable and Antidegree Equitable Graphs After studying the basic properties of antidegree equitable and independent antidegree equitable sets in a graph, in this section we give some conditions for a graph to be degree equitable. We recall the definition of degree equitable graph given by A. Anitha, 5S. Arumugam, and E. Sampathkumar [1]. Definition 4.1 Let G = (V,E) be a graph. The degree equitable graph of G, denoted by G% is defined as follows:V(G4°) = V(G) and two vertices u and v are adjacent vertices in G4 if, and only if, |deg(u) — deg(v)| < 1. Example 4.2 For any regular graph G on n vertices, we have G* = Ky. Definition 4.3 A graph H is called degree equitable graph if there exists a graph G such that He Ge, Example 4.4 Any complete graph K;, is a degree equitable graph because K, = G@° for any regular graph G on n-vertices. Theorem 4.5 Let G= (V,E) be any graph on n vertices with radius r > 3. Then (4) 1< Bo(G%) < Vn? — 4nr + 5n + 4r? — Gr. (i) Bo(G%) < [452] +1, where A = A(G) and 5 = 4(G). Proof (i) Let A be an independent set of G4° such that |A] = 69(G%°). Then A is an antidegree equitable set in G and hence 80 (G**) > degalv) = Di degelv) =D) %#-1= h(E"). vEV vEA l=1 32 Chandrashekar Adiga and K. N. Subba Krishna By Theorem 1.17 it follows that —————ee 5 ) > Bo? (G*). Therefore, 1 < Bo(G®) < Vn? — 4nr + 5n + 4r? — Gr. (ii) We know that every independent set A in G% is an antidegree equitable set in G and hence by Theorem 2.8, 1 < [OS 4 Therefore, ey < | AG) — 4(G) acat < [MARKO 4 This completes the proof. Theorem 4.6 Let H be any degree equitable graph on n vertices and H = G** for some graph G. Then A(G) — 8(@) Fasc < LO =8D where A is an independent set in G* such that |A| = Go(G%). Proof We know that if A is an independent set in H then it is an antidegree equitable set in G. Hence, Bo(H) S> degg(v) < S> 20-1= o"(H). vEA C1 By Theorem 4.5 ; S- dega(v) < (jae) + 1) : veEA | A(G) — 6(G) 2, daa) < See +1. We introduce a new concept antidegree equitable graph and present some basic results. Therefore, Definition 4.7 Let G = (V,E) be a graph. The antidegree equitable graph of G, denoted by G4 defined as follows: V(G***) = V(G) and two vertices u and v are adjacent in G2 if, and only if, |\deg(u) — deg(v)| 4 1. Example 4.8 For a complete bipartite graph Km, we have Koi if |m — n| > 2,or = 0 Ky UK, if |m—n|=1. Gade = Antidegree Equitable Sets in a Graph 33 Definition 4.9 A graph H is called an antidegree equitable graph if there exists a graph G such that H = Got, Example 4.10 Any complete graph K,, is an antidegree equitable graph because K, = G24 for any regular graph G on n-vertices. Theorem 4.11 Let G be any graph on n vertices. Then the number of edges in G%% is given by A-1 / A / n Si |S5| Si] — 2 G)- EG) ) 2 z, ©) where S; = {v| v € V dega(v) =i or i+1}, S;’ = {v| v € V dega(v) = i}, A = A(G) and 5 = 4(G). Proof By Theorem 1.16, we have the number of edges in G*“* with end vertices having the difference degree greater than two in G is A-1 A ! n [Si [Si ()- EG) +e Cr) i= 4=04+1 and also, the number of edges in G?“° with end vertices having the same degree is SY) i=6d Hence, the total number of edges in G@” is EC) £0720 i=6 i=64+1 i=od A-1 A _(n [Si| |S5"| [Si"| JEM) ALC i= i=64+1 Theorem 4.12 Let G be any graph on n vertices. Then (i) a(G2%) < n(n — 1); (ii) a(Get) < [45°] +1, where A = A(G) and 6 = 6(G). Proof Let A C V be the set of vertices that covers all edges of G2%. Then A is an antidegree equitable set in G. Hence, a(G?4?) > degg(v) = S- 22-1= a? (G4), ven f= Therefore, 2 (MA) > ren"), 34 Chandrashekar Adiga and K. N. Subba Krishna a(G?) < ./n(n — 1). Since, the set A is an antidegree equitable set in G, by Theorem 2.8, we have A-o |A| < aS +1. This implies A a(G??) < AS] +1. References [1] A. Anitha, S. Arumugam and E. Sampathkumar, Degree equitable sets in a graph, Inter- national J. Math. Combin., 3 (2009), 32-47. [2] G. Chartrand and P. Zhang, Introduction to Graph Theory, 8th edition, Tata McGraw-Hill, 2006. [3] F.Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969. International J.Math. Combin. Vol.1(2015), 35-48 A New Approach to Natural Lift Curves of The Spherical Indicatrices of Timelike Bertrand Mate of a Spacelike Curve in Minkowski 3-Space Mustafa Bilici (Department of Mathematics, Educational Faculty of Ondokuz, Mayis University, Atakum 55200, Samsun, Turkey) Evren Ergin (Cargamba Chamber of Commerce Vocational School of Ondokuz, Mayis University, Cargamba 55500, Samsun, Turkey) Mustafa Caliskan (Department of Mathematics, Faculty of Sciences of Gazi University, Teknik Okullar 06500, Ankara, Turkey) E-mail: mbilici@omu.edu.tr, eergun@omu.edu.tr, mustafacalikan@gazi.edu.tr Abstract: In this study, we present a new approach the natural lift curves for the spher- ical indicatrices of the timelike Bertrand mate of a spacelike curve on the tangent bundle T ($7) or T’ (H¢) in Minkowski 3-space and we give some new characterizations for these curves. Additionally we illustrate an example of our main results. Key Words: Bertrand curve, natural lift curve, geodesic spray, spherical indicatrix. AMS(2010): 53A35, 53B30, 53C50 §1. Introduction Bertrand curves are one of the associated curve pairs for which at the corresponding points of the curves one of the Frenet vectors of a curve coincides with the one of the Frenet vectors of the other curve. These special curves are very interesting and characterized as a kind of corresponding relation between two curves such that the curves have the common principal normal i.e., the Bertrand curve is a curve which shares the normal line with another curve. It is proved in most texts on the subject that the characteristic property of such a curve is the existence of a linear relation between the curvature and the torsion; the discussion appears as an application of the Frenet-Serret formulas. So, a circular helix is a Bertrand curve. Bertrand mates represent particular examples of offset curves [11] which are used in computer-aided design (CAD) and computer-aided manufacturing (CAM). For classical and basic treatments of Bertrand curves, we refer to [3], [6] and [12]. There are recent works about the Bertrand curves. Ekmekci and Ilarslan studied Nonnull Bertrand curves in the n-dimensional Lorentzian space. Straightforward modication of classical theory to spacelike or timelike curves in Minkowski 3-space is easily obtained, (see [1]). Izumiya 1Received August 28, 2014, Accepted February 19, 2015. 36 Mustafa Bilici, Evren Ergtin and Mustafa Caliskan and Takeuchi [16] have shown that cylindrical helices can be constructed from plane curves and Bertrand curves can be constructed from spherical curves. Also, the representation formulae for Bertrand curves were given by [8]. In differential geometry, especially the theory of space curves, the Darboux vector is the areal velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. In terms of the Frenet-Serret apparatus, the Darboux vector can be expressed as w=Tt+x«b. In addition, the concepts of the natural lift and the geodesic sprays have first been given by Thorpe (1979). On the other hand, Caliskan et al. [4] have studied the natural lift curves and the geodesic sprays in Euclidean 3-space R® . Bilici et al. [7] have proposed the natural lift curves and the geodesic sprays for the spherical indicatrices of the involute-evolute curve couple in R°. Recently, Bilici [9] adapted this problem for the spherical indicatrices of the involutes of a timelike curve in Minkowski 3-space. Kula and Yayli [17] have studied spherical images of the tangent indicatrix and binormal indicatrix of a slant helix and they have shown that the spherical images are spherical helices. In [19] Siha et. all investigated tangent and trinormal spherical images of timelike curve lying on the pseudo hyperbolic space H@ in Minkowski space-time. lyigiin [20] defined the tangent spherical image of a unit speed timelike curve lying on the on the pseudo hyperbolic space H? in R}. Senyurt and Caliskan [22] obtained arc-lengths and geodesic curvatures of the spherical indicatrices (T*) , (N*),(B*) and the fixed pole curve (C*) which are generated by Frenet trihedron and the unit Darboux vector of the timelike Bertrand mate of a spacelike curve with respect to Minkowski space R# and Lorentzian sphere S$? or hyperbolic sphere Hj. Furthermore, they give some criteria of being integral curve for the geodesic spray of the natural lift curves of this spherical indicatrices. In this study, the conditions of being integral curve for the geodesic spray of the the natural lift curves of the the spherical indicatrices (T*) , (N*),(B*) are investigated according to the relations given by [8] on the tangent bundle T ($7) or T (Hj) in Minkowski 3-space. Also, we present an example which illustrates these spherical indicatrices (Figs. 1-4). It is seen that the principal normal indicatrix (N*) is geodesic on $? and its natural lift curve is an integral curve for the geodesic spray on T (S?). §2. Preliminaries To meet the requirements in the next sections, the basic elements of the theory of curves and hypersurfaces in the Minkowski 3-space are briefly presented in this section. A more detailed information can be found in [10]. The Minkowski 3-space R} is the real vector space R°endowed with standard flat Lorentzian metric given by g = —da? + dx? + dex?, where (x1, 2,273) is a rectangular coordinate system of R? . A vector V = (v1, v2, v3) € R? is said to be timelike if g(V,V) < 0 , spacelike if g(V,V) > 0 or V = 0 and null (lightlike) if A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate 37 g(V,V) =0 and V £0 Similarly, an arbitrary f =I'(s) curve in R? can locally be timelike, spacelike or null (lightlike), if all of its velocity vectors I’ are respectively timelike, spacelike or null (lightlike), for every t € I C R. The pseudo-norm of an arbitrary vector V € R} is given by IV || = Vlg(V, V)|. T is called a unit speed curve if the velocity vector V of I satisfies ||V|| = 1. A timelike vector V_ is said to be positive (resp. negative) if and only if v; > 0 (resp.v; <0). Let [I be a unit speed spacelike curve with curvature « and torsion 7. Denote by {t(s) ,n(s),6(s)} the moving Frenet frame along the curve T in the space R?. Then t,n and b are the tangent, the principal normal and the binormal vector of the curve I, respectively. The angle between two vectors in Minkowski 3-space is defined by [21] Definition 2.1 Let X and Y be spacelike vectors in R} that span a spacelike vector subspace, then we have |g(X,Y)| < ||X||||Y]] and hence, there is a unique positive real number ~ such that |9(X, Y)| = ||XIII[¥lleosy. The real numbery is called the Lorentzian spacelike angle between X and Y. Definition 2.2 Let X and Y be spacelike vectors in R} that span a timelike vector subspace, then we have |g(X,Y)| > ||X||||Y]] and hence, there is a unique positive real number ~ such that l9(X, Y)| = ||X||I[¥lleoshy. The real number y is called the Lorentzian timelike angle between X and Y. Definition 2.3 Let X be a spacelike vector and Y a positive timelike vector in R}, then there is a unique non-negative real number yp such that \9(X,¥)| = || X|II[¥l]sinhy. The real number y is called the Lorentzian timelike angle between X and Y. Definition 2.4 Let X and Y be positive (negative) timelike vectors in R}?, then there is a unique non-negative real number yp such that 9X, Y) = ||X||||¥l|coshy. The real number y is called the Lorentzian timelike angle between X and Y. Case I. Let [ be a unit speed spacelike curve with a spacelike binormal. For these Frenet vectors, we can write TxN=-B, Nx B=-T, BxT=N where ” x” is the Lorentzian cross product in space R?. Depending on the causal character of the curve I’, the following Frenet formulae are given in [5]. T=KN, N=«T+7B,B= TN 38 Mustafa Bilici, Evren Ergtin and Mustafa Caliskan The Darboux vector for the spacelike curve with a spacelike binormal is defined by [11]: w=--TT+KB If b and w spacelike vectors that span a spacelike vector subspace then by the Definition1. we can write kK = |\w||coshy ||w|| sinh vp, 4 I where ||w||? = g (w,w) = 72 + ?. Case II. Let [ be a unit speed spacelike curve with a timelike binormal. For these Frenet vectors, we can write TxN=B, NxB=-T, BxT=-N Depending on the causal character of the curve [ , the following Frenet formulae are given in [5]. T=KkN, N=-«T4+7B,B= TN The Darboux vector for the spacelike curve with a timelike binormal is defined by [11]: w=TI-k«B There are two cases corresponding to the causal characteristic of Darboux vector w. (2) If |x| < |7| , then w is a timelike vector. In this situation, we have kK = |lw||sinhy T = |\w|| coshy, where ||w||? = —g (w, w) = 7? — K?. So the unit vector c of direction w is 1 c= —vw =sinhyT — cosh yB. Il (it) If |x| > |r|, then w is a spacelike vector. In this situation, we can write kK = |\w||coshy T = |\w||sinhg, where ||w||? = g (w, w) = «2 — r?. So the unit vector c of direction w is c = sinh yT — cosh yB. Proposition 2.5((13]) Let a be a timelike (or spacelike) curve with curvatures « and Tt. The A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate 39 curve is a general helix if and only if = is constant. Remark 2.6 We can easily see fromequations of the section Case I and Case II that: + = tan y, ~ = tanhy (or = =coth ~) , if y =constant then a is a general helix. Lemma 2.7((9]) The natural lift @ of the curve a is an integral curve of the geodesic spray X if and only if a is a geodesic on M. Definition 2.8 Let a = (a(s);T(s),N(s),B(s)) and B = (6 (s*) ;T* (s*), N* (s*) , B* (s*)) be two regular non-null curves in R?. a(s) and 3(s*) are called Bertrand curves if N(s) and N* (s*) are linearly dependent. In this situation,(a, 3) is called a Bertrand couple in R}. (See [1] for the more details in the n-dimensional space). Lemma 2.9 Let a be a spacelike curve with a timelike binormal. In this situation, G is a timelike Bertrand mate of a. The relations between the Frenet vectors of the (a, 3) is as follow L* sinh@ O coshé Wa N* |= 0 1 0 N | ,g9(T,T*) =sinhé = constant, [8]. B* cosh@ QO sinhé B Definition 2.11([10]) Let S? and H@ be hypersphere in R}?. The Lorentzian sphere and hyperbolic sphere of radius 1 in are given by S? = {V = (v1, V2, U3) E R? : g(V,V) = 1} and Hg = {V = (v1, 02,¥3) ERY: g(V,V) =—1} respectively. Definition 2.12([9]) Let M be a hypersurface in R? equipped with a metric g. Let TM be the set U{T,(M):p eM} of all tangent vectors to M. Then eachu € TM is in a unique Tp (M), and the projection x: TM — M sends v to p. Thus n~'(p) = Tp(M). There is a natural way to makeT M a manifold, called the tangent bundle of M. A vector field X € y(M) is exactly a smooth section of TM, that is, a smooth function X:M —TM such thatnoX =idy . Definition 2.13((9]) Let M be a hypersurface in R?. A curvea:I — TM _ is an integral curve of X € x(M) provided d = Xq ; that is — (a(s)) = X (a(s)) for all s € I, [10]. (1) Definition 2.14 For any parametrized curve a: I — TM, the parametrized curve given by 40 Mustafa Bilici, Evren Ergtin and Mustafa Caliskan a:Il—-~TM is called the natural lift of a on TM. Thus, we can write da d : We ae (a’ (8) lacs) = Dar(s)& (8), (3) where D is the standard connection on R}. Definition 2.15((9]) Forv€TM , the smooth vector field X € x (TM) defined by X (v) = eg (v,S(v))€ lars), € = 9 (6,8) (4) is called the geodesic spray on the manifold TM, where € is the unit normal vector field of M and S is the shape operator of M. §3. Natural Lift Curves for the Spherical Indicatrices of Spacelike-Timelike Bertrand Couple in Minkowski 3-Space In this section we investigate the natural lift curves of the spherical indicatrices of Bertrand curves (a,) as in Lemma 2.9. Furthermore, some interesting theorems about the original curve were obtained depending on the assumption that the natural lift curves should be the integral curve of the geodesic spray on the tangent bundle T (S7) or T (H@) . Note that D and D are Levi-Civita connections on S$? and H@ , respectively. Then Gauss equations are given by the followings Let D, D and D be connections in R}, S? and H@ respectively and € be a unit normal vector field of S? and Hj. Then Gauss Equations are given by the followings DxY = DxY +e9(5(X),Y)é, Dx¥ = DxY +9 (S(X),Y)ée=9 (6.8) where € is a unit normal vector field and S is the shape operator of S? (or H@). 3.1 The natural lift of the spherical indicatrix of the tangent vector of (@ Let (a,) be Bertrand curves as in Lemma 2.9. We will investigate the curve a to satisfy the condition that the natural lift curve of Gp» is an integral curve of geodesic spray, where r~ is the tangent indicatrix of 3. If the natural lift curve Bp is an integral curve of the geodesic spray, then by means of Lemma 2.9. we get, hl Bp Bre =. 0, (5) where D is the connection on the hyperbolic unit sphere H? and the equation of tangent A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate 41 indicatrix is Br» = T*. Thus from the Gauss equation we can write Dg,.br+ = Dg, bre + €9 (s (Br. ) , br) i 6S g(t F sal On the other hand, from the Lemma 2.9. straightforward computation gives dBr« ds —, ds idee = (« sinh 6 + 7 cosh 0) N a Br =tr = Moreover, we get ds 1 —_—_—_—__-_ -—- eo t x N. dsp» KsinhO+7cosh0’ kK T D at ST ———_____ 8 nee K sinh 6 + 7 coshé a kK sinh 6 + 7 coshé and g (S (tr~) ,tr«) =-—l. Using these in the Gauss equation, we immediately have K T D tr. = —-——— TT + ———_ B- nae K sinh @ + 7 coshé | anh) 7 coe foe From the Eq. (5) and Lemma 2.9.ii) we get K T pe ee —____—_eosho) B ( «& sinh é + 7 coshé sind ok (<a +rcoshO ) Since T, N, B are linearly independent, we have K T fee ee, Se et: «sinhé + 7 coshé aa «sinh 6 + 7 coshé ie It follows that, «&coshé +7 sinhé = 0 (6) T — = -—cothdé 7 ‘i co (7) So from the Eq. (7) and Remark 2.6. we can give the following proposition. Proposition 3.1 Let (a,() be Bertrand curves as in Lemma 2.9. If a is a general helix, then the tangent indicatrix Br+ of B is a geodesic on Hj. Moreover from Lemma 2.7. and Proposition 3.1 we can give the following theorem to characterize the natural lift of the tangent indicatrix of @ without proof. Theorem 3.2 Let (a,3) be Bertrand curves as in Lemma 2.9. If a is a general helix, then the natural lift Br~ of the tangent indicatrix Bp+ of 3 is an integral curve of the geodesic spray on the tangent bundle T (Hé). 3.2 The natural lift of the spherical indicatrix of the principal normal vectors of (3 Let By- be the spherical indicatrix of principal normal vectors of 3 and y+ be the natural lift 42 Mustafa Bilici, Evren Ergiin and Mustafa Caliskan of the curve . If Gy is an integral curve of the geodesic spray, then by means of Lemma 2.7. we get, Diyxtn~ = 0, (8) that is Diy» tn« = Diy. tne + eg (S (tn~),tn-) N*,e = 9(N*, N*) =1 On the other hand, from Lemma 2.9. and Case II. i) straightforward computation gives Bye =tn+ = —sinhyT + coshyB Moreover we get POOLED Fics —Ksinhy + Tcoshy ysinh y DiS a _ |W ||W'| |W B and g (S (tr) ,tn~) =1 Using these in the Gauss equation, we immediately have = pcoshy ysinh y Di,.tn~e = —-——_T + ———B. _ ||| |W Since T, N, B are linearly independent, we have pcoshyp _ psinhy _ |W |W It follows that, g=0, (9) # — = constant. (10) K So from the Eq. (10) and Remark 2.6. we can give the following proposition. Proposition 3.3 Let (a,() be Bertrand curves as in Lemma 2.9. If a is a general helix, then the principal normal indicatriz By of (3 is a geodesic on $2. Moreover from Lemma 2.7. and Proposition 4.3. we can give the following theorem to characterize the natural lift of the principal normal indicatrix of @ without proof. Theorem 3.4 Let (a,8) be Bertrand curves as in Lemma 2.9. If a is a general helix, then the natural lift BN« of the principal normal indicatrix of Bn- is B an integral curve of the geodesic spray on the tangent bundle T ($2). 3.3. The natural lift of the spherical indicatrix of the binormal vectors of (3 Let 3p be the spherical indicatrix of binormal vectors of 3 and Gg» be the natural lift of the curve 3g. . If Bg~ is an integral curve of the geodesic spray, then by means of Lemma 2.7. we get De gtae= 0, (11) A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate 43 that is Di y.tB => Dip» t B+ + eg (S$ (tg+), tp) B*,e = g (B*, B*) =1 On the other hand, from Lemma 2.9.ii) straightforward computation gives ds dsp tp~ = (Kcoshé + 7 sinhé) N Moreover we get ds 1 ee eT dsp» kcosh6+7sinhd@’ 7 9 K i D the = ———————————“T ————___ B tag «coshé + 7 sinhé e «cosh@ + 7 sinh @ and g (S (tp) ; tp) =-l. Using these in the Gauss equation, we immediately have K ce pated «cosh@ +7 sinhdé «.cosh@é + 7 sinh@ + B* From the Eq. (11) and Lemma 2.9.ii) we get a T ~ x cosh + T sinh —_________ + sinh@} B = 0. ( gg toon) P+ (a sin ) 0 Since T, N, B are linearly independent, we have K -—e h = pee orem y ® % : PTE ear ad eee it follows that «sinh @ + 7 coshé = 0 (12) we — =—tanhé 13 : an (13) So from the Eq. (13) and Remark 2.6. we can give the following proposition. Proposition 3.5 Let (a,() be Bertrand curves as in Lemma 2.9. If a is a general helix, then the binormal indicatriz 3p« of 3 is a geodesic on S?. Moreover from Lemma 2.7. and Proposition 4.5. we can give the following theorem to characterize the natural lift of the binormal indicatrix of @ without proof. Theorem 3.6 Let (a, 3) be Bertrand curves as in Lemma 2.9. If a is a general helix, then the natural lift Bg» of the binormal indicatrit Bp of 3 is an integral curve of the geodesic spray on the tangent bundle T (Sr) From the classification of all W-curves (i.e. a curves for which a curvature and a torsion are constants) in (Walrawe, 1995), we have following proposition with relation to curve. 44 Mustafa Bilici, Evren Ergiin and Mustafa Caliskan Proposition 3.7 (1) If the curve a with K =constant > 0,7 =0 then a is a part of a circle; (2) If the curve a with & =constant > 0 , 7 =constant #0 , and |r| > & then a is a part of a spacelike hyperbolic helix, a(s)=2 («sinh (Vs) ,V72Ks, kcosh (Vs) ) , K=7?—#?; (3) If the curve a with « =constant > 0, rT =constant 4 0 and |r| < &, then a is a part of a spacelike circular helix, a(s)=2 (V7?Ks, «cos (Vs) sin (Vs) VP Ks, ) ,K=K*-7’; From Lemma 3.1 in Choi et al 2012, we can write the following proposition. Proposition 3.8 There is no spacelike general helix of spacelike curve with a timelike binormal in Minkowski 3-space with condition |r| = |K]. Example 3.9 Let a(s) =} (sinh (Vs) , 2\/3s, cosh (3s) )be a unit speed spacelike hyperbolic helix with 7 = “(cosh (3s) ,2, sinh (V3s)) N = (sinh (v3s) , 0, cosh (v3s)) ; k=landt=2 B= a (2 cosh (v3s) , 1,2 sinh (v3s)) In this situation, spacelike with spacelike binormal Bertrand mate for can be given by the equation — 2v3 (a+ 4) ams (Vi) eR For \= % B(s)= (4 sinh (v3s) as i cosh (ve) ) : The straight forward calculations give the following spherical indicatrices and natural lift 45 A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate curves of spherical indicatrices for (, Bbr~ = “ (v7cosh (v3s) 2, /7sinh (v3s)) Bnx = (sinh (v3s) , 0, cosh (v3s)) Bax = _ (-2 cosh (v3s) : i, —2sinh (v%)) Bre = : (v7 sinh (v3s) , 0, V7 cosh (v3s)) Bnx = (cosh (v3s) , 0, sinh (v3s)) Bp» = —2 (sinh (v3s) , 0, cosh (v3s)) respectively, (Figs. 1-4). Figure 1. Tangent indicatrix 37. for Bertrand mate of a on H 46 Mustafa Bilici, Evren Ergiin and Mustafa Caligkan UT pl s $ W717 tt 3) ay vA IE Ge =e ES... i it fi Fh ANAS OT; SS | ED I, | bit 5 mimi My 7 gf 8 jaaees \S rT =r & y. = “—e ze manatee! Hs OSES SASL CORSE = Figure 3. Binormal indicatrix 3+ for Bertrand mate of a on S$? A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate 47 ewe: EE tT j j ; i O -\ wth Dae itt | ' ! | | | | 7 ;2 ) py } ! Ai ja | Figure 4. Principal norma indicatrix Gy~ and its natural lift curve By- on S$? References 1 me "oo aw) Ou Ekmekci N., Ilarslan K., On Bertrand curves and their characterization, Differential Ge- omety Dynamical Systems, 3, No.2, 17-24, 2001. Thorpe J.A., Elemantary Topics In Differantial Geometry, Springer-Verlag, New York, Heidelberg-Berlin, 1979. M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Pearson Education, 1976. Caliskan M., Sivridag A.J. & Hacisalihoglu H.H., Some characterizations for the natural lift curves and the geodesic spray, Commun. Fac. Sci. Univ., 33: 235-242, 1984. Petrovic-Torgasev M., Sucurovic E., Some characterizations of the spacelike, the timelike and null curves on the pseudohyperbolic space in ??, Kragujevac J. Math., 22: 71-82, 2000. W. Kuhnel, Differential Geometry: Curves-Surfaces-Manifolds, Braunschweig, Wiesbaden, 1999. Bilici M., Cahskan M. & Aydemir I., The natural lift curves and the geodesic sprays for the spherical indicatrices of the pair of evolute-involute curves, Int. J. of Appl. Math., 11(4): 415-420, 2003 Oztekin H. B., Bektas M., Representation formulae for Bertrand curves in the Minkowski 3-space, Scientia Magna, 6(11): 89-96, 2010. Bilici M., Natural lift curves and the geodesic sprays for the spherical indicatrices of the involutes of a timelike curve in Minkowski 3-space, International Journal of the Physical 48 10 11 12 13 14 15 16 17 18 19 Mustafa Bilici, Evren Ergiin and Mustafa Caliskan Sciences, 6(20): 4706-4711, 2011. O'Neill B., Semi Riemann Geometry, Academic Press, New York, London, 1983. Ugurlu H.H., On the geometry of time-like surfaces, Commun. Fac. Sci. Univ. Ank., Series Al, 46: 211-223, 1997. Struik D.J., Differential Geometry, Second ed., Addison-Wesley, Reading, Massachusetts, 1961. Choi J.H., Kim Y.H. & Ali A.T., Some associated curves of Frenet non-lightlike curves in ??, J. Math. Anal. Appl., 394: 712-723, 2012. Barros M., Ferrandez A., Lucas P. & Merono M.A., General helices in the three-dimensional Lorentzian space forms, Rocky Mountain J. Math., 31(2): 373-388, 2001. Nutbourne A.W., Martin R. R., —it Differential Geometry Applied to the Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988. Izumiya S., Takeuchi N., Generic properties of helices and Bertrand curves, J. Geom., 74: 97-109, 2002. Kula L., YayliY., On slant helix and its spherical indicatrix, Applied Mathematics and Computation, 169(1): 600-607, 2005. Millman R.S., Parker G.D., Elements of Differential Geometry, Prentice-Hall Inc., Engle- wood Cliffs, New Jersey, 1977. Yilmaz S., Ozyilmaz E., YaylY. & Turgut M., Tangent and trinormal spherical images of a time-like curve on the pseudohyperbolic space, Proc. Est. Acad. Sci., 59(3):216-224, 2010. [20] lyigiin E., The tangent spherical image and ccr-curve of a time-like curve in ??, 2013. [21] Journal of Inequalities and Applicat2ions,10.1186/1029-242X-2013-55. [22] Ratcliffe J.G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, Inc., New York, 1994. [23] Senyurt S., Caliskan O.F., The natural lift curves and geodesic curvatures of the spherical indicatrices of the spacelike-timelike Bertrand curve pair, International J. Math. Combin., 2: 47-62, 2014. International J.Math. Combin. Vol.1(2015), 49-56 Totally Umbilical Hemislant Submanifolds of Lorentzian (a)-Sasakian Manifold Barnali Laha and Arindam Bhattacharyya (Department of Mathematics, Jadavpur University, Kolkata, India) E-mail: barnali.laha87Q@gmail.com, bhattachar1968@yahoo.co.in Abstract: This paper is summarized as follows. In the first section we have given a brief history about slant and hemi-slant submanifold of Lorentzian (a@)-Sasakian manifold. This section is followed by some preliminaries about Lorentzian (a)-Sasakian manifold. Finally, we have derived some interesting results on the existence of extrinsic sphere for totally umbilical hemi-slant submanifold of Lorentzian (a@)-Sasakian manifold. Key Words: Totally Umbilical, hemi-slant submanifold, extrinsic sphere. AMS(2010): 53025, 53C40, 53C42, 53D15 §1. Introduction Chen in 1990 [2] initiated the study of slant submanifold of an almost Hermitian manifold as a natural generalization of both holomorphic and totally real submanifolds. After this many research papers on slant submanifolds appeared. The notion of slant immersion of a Riemannian manifold into an almost contact metric manifold was introduced by A. Lotta in 1996 [5]. He studied the intrinsic geometry of 3-dimensional non-anti-invariant slant submanifolds of K- contact manifold. Further investigation regarding slant submanifolds of a Sasakian manifold [8] was done by Cabrerizo et al. in 2000. Khan et al. in 2010 defined and studied slant submanifolds in Lorentzian almost paracontact manifolds [14]. The idea of hemislant submanifold was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds in [9]. Recently, in 2009 totally umbilical slant submanifolds of Kaehler manifold was studied by B.Sahin. Later on, in 2011 Siraj Uddin et.al. studied totally umbilical proper slant and hemislant submanifolds of an LP-cosymplectic manifold [21]. Our present note deals with a special kind of manifold i.e. Lorentzian (a)-Sasakian man- ifold. At first we give some introduction about the development of such manifold. An almost contact metric structure (¢, €,17, g) on M is called a trans-Sasakian structure [17] if (MX R, J, G) belongs to the class W4 [11], where J is the almost complex structure on (VX R) defined by d d 5) = (6X =f )5) be (J, 7 1Received May 23, 2014, Accepted February 20, 2015. 50 Barnali Laha and Arindam Bhattacharyya for all vector fields X on M and smooth functions f on M x R , G is the product metric on MXR. This may be expressed by the condition (Vx@)¥ = alg(X, Y)E + (VY) X] + Blg(OX,Y) — n(V) bX], for some smooth functions a and 3 on M in [1], and we say that the trans-Sasakian structure is of type (a, 8). A trans-Sasakian structure of type (a, 3) is a-Sasakian, if G = 0 andaa nonzero constant [13]. If a = 1, then a-Sasakian manifold is a Sasakian manifold. Also in 2008 and 2009 many scientists have extended the study to Lorentzian (a)-Sasakian manifold in [22], [18]. In this paper we have studied some special properties of totally umbilical hemislant submanifolds of Lorentzian (a)-Sasakian manifold. §2. Preliminaries An n-dimensional Lorentzian manifold M is a smooth connected paracontact Hausdorff mani- fold with a Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type (0,2) such that for each point p € M, the tensor g, : T,M x T,M -— R is a non-degenerate inner product of signature (—,+,+,:--,+), where T,M denotes the tangent vector space of M at p and R is the real number space. A non-zero vector v € T,M is said to be timelike if it satisfies g,(v,v) < 0 [16]. Let M be an n-dimensional differentiable manifold. An almost paracontact structure (¢,€,7,g), where ¢ is a tensor of type (1,1), € is a vector field, 7 is a 1-form and g is Lorentzian metric, satisfying following properties : PX =X+7fX)E, nod=0, ¢=0, n(€) =-1, (2.1) G(X, OY) = g( X,Y) + (X)n(V), g(X, €) = 0(X). (2.2) for all vector fields X,Y on M. On M if the following additional condition hold for any MV erm, (Vx@)¥ = alg(X, Y)E + n(V)X], (2.3) Vx = aX, (2.4) where V is the Levi-Civita connection on M, then M is said to be an Lorentzian a-Sasakian manifold (Matsumoto, 1989 [15], [22]). Let M be a submanifold of M with Lorentzian almost paracontact structure (@,€, 1,9) with induced metric g and let V is the induced connection on the tangent bundle 7M and V+ is the induced connection on the normal bundle T+M of M. The Gauss and Weingarten formulae are characterized by VxY =VxY +A(X,Y), (2.5) VxN =—-AyX + VEN, (2.6) for any X,Y € TM, N € T+M, his the second fundamental form and Ay is the Weingarten Totally Umbilical Hemislant Submanifolds of Lorentzian (a)-Sasakian Manifold 51 map associated with N via For any X € T'(TM) we can write, oX =TX + FX, (2.8) where TX is the tangential component and FX is the normal component of ¢X. Similarly for any N € 1(T+M) we can put oV =tV + fV, (2.9) where tV denote the tangential component and fV denote the normal component of éV. The covariant derivatives of the tensor fields T and F are defined as (Vxd)¥ =VxdY —oVxY V X,YeETM, (2.10) (VxT)Y =VxTY —TVxY V X,YeETM, (2.11) (VxF)Y =ViFY —FVxY, V X,YeETM. (2.12) From equation (2.3), (2.5), (2.8), (2.9), (2.11) and (2.12) we can calculate (VxT)Y = alg(X,Y)E + (VY) X] + Ary X + th(X,Y), (2.13) (VxF)Y =—A(X,TY) + fh(X,Y). (2.14) A submanifold M is said to be invariant if F' is identically zero, ie., 6X €T(TM) for any X €T(TM). On the other hand, M is said to be anti-invariant if T is identically zero, i-e., @X €1(T+M) for any X €T(TM). A submanifold M of M is called totally umbilical if h(X,Y) = 9(X,Y)H, (2.15) for any X,Y € T (7M). The mean curvature vector H is denoted by H= eo h(e;,e;), where k is the dimension of M and {e1, e2,e€3,--- , ex} is the local orthonormal frame on M. A submanifold M is said to be totally geodesic if h(X,Y) = 0 for each X,Y € I(7M) and is minimal if H = 0 on M. §3. Slant Submanifolds of a Lorentzian (alpha)-Sasakian Manifold Here, we consider M as a proper slant submanifold of a Lorentzian (a)-Sasakian manifold M. We always consider such submanifold tangent to the structure vector field €. Definition 3.1 A submanifold M of M is said to be slant submanifold if for any x € M and X €T,M \ €, the angle between 6X and T,M is constant. The constant angle 6 € [0,7/2] is then called slant angle of M in M. If 0 = 0 the submanifold is invariant submanifold, if 0 = m/2 52 Barnali Laha and Arindam Bhattacharyya then tt is anti-invariant submanifold and if 0 4 0,72/2 then it is proper slant submanifold. From [20] we have Theorem 3.1 Let M be a submanifold of an Lorentzian (a)-Sasakian manifold M such that €€TM. Then M is slant submanifold if and only if there exists a constant » € [0,1] such that T? =I +n@&). (3.1) Again, if @ is slant angle of M, then \ = cos? @. From [20], for any X,Y tangent to M, we can easily draw the following results for an Lorentzian (a)-Sasakian manifold M, (TX, TY) = cos*O{g(X,¥Y) +(X)n(¥)}, g(PX, FY) = sin?0{g(X,Y) + n(X)n¥)}- Definition 3.2 A submanifold M of M is said to be hemi-slant submanifold of a Lorentzian (a)-Sasakian manifold M if there exists two orthogonal distribution D, and Dz on M such that (a) TM =D, @®D2® < E>; (b) The distribution D, is anti-invariant i.e., 6D, CT+M; (c) The distribution Dz is slant with slant angle 6 4 1/2. If us is invariant subspace under @ of the normal bundle T+ M, then in the case of hemi-slant submanifold, the normal bundle T--M decomposes as T+M =< p> ObD* © FDo. The curvature tensor of an Lorentzian (a:)-Sasakian manifold is defined as [4] R(X, Y)Z =VxVyZ -—VyVxZ — VixyyZ. (3.2) For the curvature tensor we can compute by using the equations (2.10) and (3.2) the relation R(X,Y)$Z = oR(X,Y)Z+079(Y, Z)OX — a7 Q(X, Z)bY (3.3) —a’g([X,Y], Z)6X + ag(X, Vy Z)E + an(VyZ)X —ag(Y, VxZ)E— an(VxZ)Y¥ — an(Z)VxY tan(Z)Vy X — an(Z)[X,Y] +ag(VxY, Z)E + ag(Vy X, Z)E. Definition 3.3 A submanifold of an arbitrary Lorentzian (a)-Sasakian manifold which is totally umbilical and has a nonzero parallel mean curvature vector [10] is called an Extrinsic sphere. §4. Main Results This section mainly deals with a special class of hemi-slant submanifolds which are totally Totally Umbilical Hemislant Submanifolds of Lorentzian (a)-Sasakian Manifold 53 umbilical. Throughout this section we have considered M as a totally umbilical hemi-slant submanifold of Lorentzian (a)-Sasakian manifold. We derive the following. Theorem 4.1 Let M be a totally umbilical hemi-slant submanifold of a Lorentzian (a)-Sasakian manifold M such that the mean curvature vector H €< pu >. Then one of the following is true: (i) M is totally geodesic; (it) M is semi-invariant submanifold. Proof For V € @¢D+ and X € Do, we have from (2.3), (2.5),(2.6) and (2.10) alg(X, VE + n(V)X] = VxeV + g(X, OV) H + AVX — OVRV. (4.1) Since the distributions are orthogonal and from the assumption that H © p, above equation can be written as (VxV, H) = g(V,VxH) = 0. (4.2) This implies V{H € pS FDo. Now for any X € Do, we obtain on using the Gauss and Weingarten equations alg(X, HE + n(H)X] = VxeH — Agu X + bAnX — OVEH. (4.3) Now, using the assumption that , M is totally umbilical we have an(H)X = VxoH — Xg(H, 6H) + oXg(H, H) — 6VXH. (4.4) On using equation (2.8) we calculate an(H)X = VxoH+TXg(H, H) + FXg(H, H) — oVEH. (4.5) Taking inner product with FX € FDs, an(H)g(X, FX) = g(VxoH, FX) + g(FX, FX)9(H, H) — 9(6V¥H, FX). (4.6) From Theorem 3.1 the equation becomes an(H)g(X, FX) — g(VxoH, FX) — sin*6|\H|[?||X|/? + 9(OVXH, FX) = 0. (4.7) If either H A 0 then Dg = {0}, ie. M is totally real submanifold, and if Dg 4 {0}, M is totally geodesic submanifold or M is semi-invariant submanifold. For any Z € D+ from (2.13) we get ValZ—TVzZ =alg(Z, Z)E+n(Z)Z] + ApzZ + th(Z, Z). (4.8) 54 Barnali Laha and Arindam Bhattacharyya Taking inner product with W € D+ the above equation takes the form g(VzTZ,W)-g(TVzZ,W) = alg(Z,Z)9(€,W) +7(Z)g(Z,W)] (4.9) +g(ArzZ, W) + g(th(Z, Z), W). As M is totally umbilical hemi-slant submanifold and using (2.7) we can write WVzTZ,W) — 9(TVzZ, Z) = a9(Z, W)g(H, FZ) + o(tH, W)||Z||?. (4.10) The above equation has a solution if either H € y or dim D+ = 1. If however, H does not belong to yz then we give the next theorem. Theorem 4.2 Let M be a totally umbilical hemi-slant submanifold of a Lorentzian (a)-Sasakian manifold M such that the dimension of slant distribution Dg > 4 and F is parallel to the submanifold, then M is either extrinsic sphere or anti-invariant submanifold. Proof Since the dimension of slant distribution Dg > 4, therefore we can select a set of orthogonal vectors X,Y € Dg, such that g(X,Y) = 0. Now by replacing Z by TY in (3.4) we have for any X,Y,Z € Do, R(X,Y)¢TY = @R(X,Y)TY +07g(¥, TY)OX (4.11) —a’g(X,TY)bY — a79([X, Y], TY) tag(X, Vy TY )é + an(VyTY)X —ag(Y, VxTY)é — an(VxTY)Y. Now using equation (2.3) and (3.1) we obtain on calculation R(X, Y)FTY + cos?0R(X,Y)Y = @R(X,Y)TY +07g(¥,TY)bX (4.12) —a’g(X,TY)bY — a79([X, Y], TY) tag(X,VyTY )é + an(VyTY)X —ag(Y, VxTY)é — an(VxTY)Y. Again if F is parallel, then above equation can be written as FR(X,Y)TY + cos?@R(X,Y)Y = @R(X,Y)TY +079(Y,TY)OX (4.13) —a’g(X,TY)bY — a79([X,Y], TY) +ag(X, Vy TY )é + an(VyTY)X —ag(Y, VxTY)é — an(VxTY)Y. Taking inner product with N ¢ T+M, we obtain on using (3.3) and the orthogonality of X and Y vectors, cos*6||¥ ||?9(VkH,N) = 0 Totally Umbilical Hemislant Submanifolds of Lorentzian (a)-Sasakian Manifold 55 The above equation has a solution if either 9 = 7/2 ie. M is anti-invariant or Vy H = 0V X € Dg. Similarly for any X € D+@ < € > we can obtain VEH = 0, therefore V}H = 0V X € TM i.e. the mean curvature vector H is parallel to submanifold, i.e., M is extrinsic sphere. Hence the theorem is proved. Now we are in a position to draw our main conclusions following. Theorem 4.3 Let M be a totally umbilical hemi-slant submanifold of a Lorentzian (a)-Sasakian manifold M. then M is either totally geodesic, or semi-invariant, or dim D+ = 1, or Extrinsic sphere, and the case (iv) holds if F is parallel and dim M > 5. Proof The proof follows immediately from Theorems 4.1 and 4.2. References 1 [12 [13] D.E.Blair and J.A.Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publications Matematiques, Vol. 34, 1990, pp. 99-207. B.Y.Chen, Slant immersions, Bulletin of the Australian Mathematical Society, vol. 41, no.1, pp. 135-147, 1990. 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A.Gray and L.M.Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali di Matematica Pura ed Applicata, Vol. 123, No. 4, 1980, pp. 35-58. doi:10.1007/BF01796539. S.K.Hui, 5.Uddin, C.Ozel, A.A.Mustafa, Warped product submanifolds of LP-Sasakian manifold, Hindawi Publishing Corporation, Discrete Dynamics in nature and Society, Vol.2012, Article ID 868549. D.Janssens and L.Vanhecke, Almost contact structures and curvature tensors, Kodai Math- ematical Journal, Vol. 4, No. 1, 1981, pp.1-27. doi:10.2996/kmj/1138036310. 56 14 15 16 17 18 19 20 (21 [22 Barnali Laha and Arindam Bhattacharyya M.A.Khan, K.Singh and V.A.Khan, Slant submanifolds of LP-contact manifolds, Differ- ential Geometry Dynamical Systems, vol.12, pp. 102-108, 2010. K.Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Sct., 12, pp. 151-156. B.O’Neill, Semi Riemannian Geometry with Applications to Relativity, Academy Press, Inc.1953. J.A.Oubina, New classes of contact metric structures, Publicationes Mathematicae Debre- cen, Vol.32, No. 4, 1985, pp. 187-193. D.G.Prakasha, C.S.Bagewadi and N.S.Basavarajappa, On pseudosymmetric Lorentzian a- Sasakian manifolds, IJPAM, Vol. 48, No. 1, 2008, 57-65. B.Sahin, Every totally umbilical proper slant submanifold of a Kaehler manifold is totally geodesic, Results in Mathematics, vol.54, no.1-2, pp. 167-172, 2009. Khushwant Singh, Siraj Uddin, Cenap Ozel, M.A.Khan, A class of totally umbilical slant submanifolds of Lorentzian para-saskian manifolds, International Journal of Physical Sci- ence, vol. 7(10), pp. 1526-1529, 2012. S.Uddin, M.A.Khan, K.Singh, Totally Umbilical Proper slant and hemislant submanifolds of an LP-cosymplectic manifold, Hindawi Publishing Corporation, Mathematical Problems in Engineering, vol.2011, Article ID 516238. A.Yildiz, M.Turan and B.F.Acet, On three dimensional Lorentzian a-Sasakian manifolds, Bulletin of mathematical Analysis and Applications, Vol. 1, Issue 3(2009), pp. 90-98. International J.Math. Combin. Vol.1(2015), 57-64 On Translational Hull Of Completely 7*~-Simple Semigroups Yizhi Chen', Siyan Lit and Wei Chen? 1. Department of Mathematics, Huizhou University, Huizhou 516007, P.R.China 2. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, P.R.China. E-mail: yizhichen1980@126.com Abstract: In this paper, we give a construction theorem about the translational hull of completely 7*’*~—simple semigroups which extends the translational hulls of completely J*—simple semigroups and completely simple semigroups. Key Words: Translational hull; completely 7*’~—simple semigroup; construction. AMS(2010): 20M10 §1. Introduction Let S be a semigroup. A mapping » from a semigroup S to itself is a left translation of S' if (ab) = (Aa)b for all elements a,b of S; a mapping p from 5 to itself is a right translation of S if (ab)p = a(bp) for all elements a,b of S. A left translation \ and right translation p are linked if a(Ab) = (ap)b for all a, b of S, in this case, the pair (A, p) is a bitranslation of S. The set A(S) of all left translations of S and the set P(S) of all right translations of S' are semigroups under the composition of mappings. The translational hull of $ is the subsemigroup 2(S) of A(S) x P(S) of all bitranslations of S. A left translation \ is inner if \ = Aq for some a € S, where \gx2 = ax for all x € S; an inner right translation pq is defined dually; the pair 7, = (Aa, Pa) is an inner bitranslation and the set II(.S) of all inner bitranslations is the inner part of Q(S) (actually an ideal of Q(S)). The translation hull of semigroups plays an important role in the algebraic theory of semigroups. It is an important tool in the study of ideal extensions. For more related details of translational hulls, the reader is referred to [1], [4], [5], [7],[14],[15].. In order to generalize regular semigroups, new Green’s relations, namely, the Green’s *- relations on a semigroup have been introduced as follows ({11], [12]): Le={(a,b)e SxS: (Vz,y € S')ax = ay & ba = by}, R* ={(a,b)E SxS: (Vz,y € S')ra = ya & rb = yd}, 1Supported by Grants of the NNSF of China (No. 11401246, 11426112); the NSF of Guangdong Province (Nos.2014A030310087, 2014A030310119); the NSF of Fujian Province(2014J01019); the Outstanding Young Innovative Talent Training Project in Guangdong Universities (No. 2013LYM0086); Science Technology Project of Huizhou City . 2Received May 23, 2014, Accepted December 2, 2014. 58 Yizhi Chen, Siyan Li and Wei Chen H= LIAR", Di=LVR*, (a,b) € J* = J*(a) = J*(b), where J*(a) and J*(b) are the principal x— ideals generated by a and b respectively. In [3], Fountain investigated a class of semigroups called abundant semigroups in which each £*—class and each R*—class of S contain at least an idempotent. And from which, we know that, the class of regular semigroups are properly contained in the one of abundant semigroups. According to [3], a semigroup in which every idempotent is primitive is said to be a primitive semigroup, and an abundant semigroup S is called a completely 7*—simple semigroup if S$ itself is primitive and the idempotents of S' generate a regular subsemigroup of S. Clearly, completely J*—simple semigroups extend completely simple semigroups studied by Clifford and Petrich in [2]. Later on, Ren and Shum [16] investigated the structure of superabundant semigroups, and generalized the corresponding results of completely regular semigroups in [15]. On the other hand, in order to further generalize completely regular semigroups [super- abundant semigroups] in the class of rpp semigroups, Guo, Shum and Gong [10] introduced the so-called (*,~)-Green’s relations on a semigroup S. The relations £*~ and R*~ are respec- tively defined as £* and R. The intersection and the join of £L*~ and R*~ are respectively denoted by H*’~ and D*’~ . The relation 7*’~ is defined by the rule that a7*’~b if and only if J*~(a) = J*~(b), Where, for any a,b € S, aRb if and only if for all e € E(S),ea = a if and only if eb = b, and J**~(a) is the smallest ideal containing a and saturated by £L**~ and R*~. According to [10], a semigroup S is called an r-ample semigroup if S is £**~-abundant and R*~-abundant, here we call that S is c-abundant, if every equivalence o-class of S contains idempotents of S. An r-ample semigroup is called a super-r-ample semigroup, if S is H*’~- abundant. The class of super-r-ample semigroups forms a proper extension class of the class of superabundant semigroups. It was shown in [10] that R*’~ usually is not a left congruence on S even if S is anR*’~-abundant semigroup, but in a super-r-ample semigroup S, the relation R*’~ is a left congruence on S. In [9], the authors defined a class of completely .7**~-simple semigroups, and give the structure of such semigroups which extended the celebrated Rees theorem for completely simple semigroups. According to [9], a super-r-ample semigroup S is called a completely 7**~-simple semigroup if S is 7**~-simple. Clearly, a completely 7*-simple semigroup must be completely J**~-simple. Note from [15] and [1] that, the translational hulls of completely simple semigroups and completely 7*—simple semigroups have been solved, so naturally, we will quote such a question: what is the translational hull of completely 7**~—simple semigroups, do we have some similar results with the ones of completely 7**~—simple semigroups or completely simple semigroups? In this paper, we will set out to discuss the above question, and finally establish a construc- tion theorem about the translational hull of completely 7**~—simple semigroups which extend the translational hulls of completely 7*—simple semigroups and completely simple semigroups. On Translational Hull Of Completely 7*’~-Simple Semigroups 59 For notations and terminologies not mentioned in this paper, the readers are referred to [5],{9] or [10]. §2. Main Results Definition 2.1 ((6], Definition 1) Let M[T;I,A;P] be a Rees matrix semigroup and P the Ax I matriaz over a left cancellative monoid T. Then P is said to be normalized at 1 if there is an element 1é€ INA such that py; = pyi =e for allie I, X€ A, where e is the identity of the left cancellative monoid T. Furthermore, the Rees matrix semigroup M[T; I, A; P] is called normalized if P is normalized. Lemma 2.2 ((6], Theorem 1) Let T be a left cancellative monoid with an identity element e and I, A be nonempty sets. Let P = (pyi) be aA x I matrix where each entry in P is a unit of T. Suppose that P is normalized at1eE1IMA. Then the normalized Rees matrix semigroup M = M[T;I,A;P] is completely J*~—simple semigroup. Conversely, every completely J**~—simple semigroup is isomorphic to a normalized Rees matrix semigroup M = M|T;I,A;P] over a left cancellative monoid T. By Lemma 2.2, we know that if S' is a completely 7*’~— simple semigroup, then it can be isomorphic to a normalized Rees matrix semigroup M = M[T; I, A; P] over a left cancellative monoid T. Hence, to discuss the translational hulls of completely 7**~—simple semigroups, we can also consider the cases of normalized Rees matrix semigroups M = M[T;I,A;P] over a left cancellative monoid T for convenience. In the following, we will establish the translational hull of a normalized Rees matrix semi- group M = M[T;I, A; P] over a left cancellative monoid T. Before we give our main result, it will be useful to make use of the following notation. Notation We set M[T;J,A;P] with P normalized at 1 € I x A and denoted by e the identity of T. Let T(S) = {(F,t,®) €T (I)xTxT(A)| for all i € I, w € A, py ritpies = Pu,Fitppe,i} with multiplication (F,t,0)(F’,t',®) = (FF, tpyg,p/it ,®®) for all i € I and \ € A, where T’ (I) (T(A)) means the semigroup of all full transformations on I (A) and all of the transformations are written on the left (right). Theorem 2.3 Let S = M[T;I,A;P] with P normalized, each entry in P is a unit of T,and let e be the identity of T. Define a mapping o from Q(S) to T(S) by a: (A,p) > (Ft.®) ((A,p) € (5) where F,t and ® are defined by the requirements A(t; 6, l= (Pies gee) C8, (1) Ce, p= Go, 822+) 68) (2) 60 Yizhi Chen, Siyan Li and Wei Chen (lemwp=(-+,-++ wee S. (3) Further define a mapping t from T(S) to Q(S) by 7: (F,t,®) > (Ap) ((F,t,®) € T(S)), where and p are defined by the formulas Ai, hw) = (Fi,tpiesh,w) ((t,h,m) € 8), (4) (i,h, w)p = (i, hpp.rit, w®) ((i,h,w) € S). (5) Then o and Tt are mutually inverse isomorphisms between Q(S) and T(S). Proof We will show the theorem by the following steps. (i) o is a mapping. Let (A, p) € Q(S). For any (7, h, w) € S', we have Ai, h, w) = AG, h, (1, e, w)] = AG, A, 1), e, 4), so that A(i,h, 2) = (j,h', 2) for some j € I and h’ € T. Similarly, we have (i, h, )p = (i, hh”, v) for some h” € T andy € A. In the following, we will use the above statements repeatedly. In particular, we may define s; and r, by A(i,e, 1) = (Fi, s:,1) (Get), (le,wp=(1,ryu,w®) (A€ A). By the definition of t in this theorem, we have t = r;. Also, notice that [(1, €, 1)l(é, ¢, 1) = (1,t,18)(é, 6,1) = (1, tria,i, 1), (1, e, 1)[A(@, e, 1)] = (1, e,1)(F%, s;,1) = (1, s:, 1), we have s; = tpio,;. Thus, Xi, h, Lt) oo ALG, €, 1)d, h, L)| = [A(i, €, 1)JQ, h, Lt) = (Fi, s:,1)(1, A, w) ie (Fi, tpie,,1)(1, A, pw) = (Fi, tpia ih, p). This proves (4). With a similar argument, we can establish (5). Hence, (is €, L)[AG, e, 1)] = (1, e, pw) (Fi, tpiei, 1) = (1, pu Fitpio., Ls (1, €, LL) p| (4, e, 1) = (1, Puri; UP) (4, e, 1) = (1, pu, FitPps,i, De On Translational Hull Of Completely 7*’~-Simple Semigroups 61 Since (A, p) € 0(S), we have p, ritpie,; = Py,Fitppe i, and then (F,t,®) € T(S), and (2) holds. (ii) 7 is a mapping. Let (F,t,®) € T(S), and let A and p be defined as (4) and (5) respectively. Then for (i,h, uw), (9k, v) € S, we have [A(i, h, mW); Ki 7) I (Fi, tpia ih, w)(9, kv) = (Fi, tpie shpyjk, v) A(i, hp k, v) = A[(i,h, WG, k,v)). l| Hence, A is a left translation. Similarly, we can show that p is a right translation. Further, on the one hand, (i, h, 1) [A(j, k, v)| = (i, h, uw) (FJ, tie jk, Vv) = (i, hpyFytpiojk,v), (6) on the other hand, [(Z, h, LL) p| ee k, v) _ (i, hpyFit, UP) (J, k, v) = (4, hpy Fitpps,jk, v), (7) and notice that (F,t,®) € T(S), we can immediately obtain that (6) and (7) are equal. And then (A, p) € Q(S). Thus, 7 is a mapping from T(S') to (5), and (zi) holds. (iit) or is an identity mapping on 2(S). Let (A, p) € Q(S), and let (A, p)or = (F,t,®)r = (X',p ) so that (i, h, w) = (Fi, tpis sh, 11). By the proof of (i), we know (4) holds for A,thus, we have \ = \’. Similarly, we have p = p . Hence, (iti) holds. (iv) ro is an identity mapping on T(S). Let (F,t,®) € T(S), and let (F,t,®)ro = (A,p)o = (F’,t,®'). Then (4) and (5) are satisfied, and thus A(i,e,1) = (Fi, tpie.,1), (le, u)e = (1, Pu, rit, u®). By (1),(2) and (3), we immediately obtain that F = F = = and @= @. Consequently, To is the identity mapping on T(S). (v) 7 is a homomorphism. Let (F,t,®)r = (A,p), (Ft, ® )r =(X',p), and (FF' , tpg pt ,®® )r = (€,n). On the one hand, we have MD (i,k, p) = AF i,t pie wh, u) = (FF i, tpg. pit Pie’ hs W). (8) On the other hand , E(t, h, pw) = (PF 2, trio rit Proo’ its ML): (9) Since (F’,t',®) € T(S), we have Dis Let Digs = Dis Alit Dees and then (8) and (9) are equal. That is, AV = €. Similarly, we can prove that pp =. Therefore, 7 is a homomorphism. 62 Yizhi Chen, Siyan Li and Wei Chen (vt) Analogous with the proof of (v), we can prove that o is a homomorphism. Summing up the six steps above, we have shown that both o and 7 are isomorphisms. Remark 2.4 From Theorem 2.3, we know that, under the isomorphism, the translational hull of a normalized Rees matrix semigroup M = M[T; I, A; P] over a left cancellative monoid T can regard as the semigroup T(S), whose elements and multiplications are defined in the Notation. And then by Lemma 2.2, the translational hull of a completely 7*’~-simple semigroup can be also regarded as this form up to isomorphism. * Further, from Remark 1 in [9], we know that if S$ is an abundant semigroup, then R**~=R*. Hence S is a completely 7**~ -simple semigroup if and only if S is a completely 7*-simple semigroup; S is a left cancellative monoid if and only if S is a cancellative monoid. If S is a regular semigroup, then R**~=R*, £L**~=L*. Hence S is a completely 7*’~-simple semigroup if and only if S is a completely simple semigroup; S is a left cancellative monoid if and only if S is a group. Now, if we let left cancellative monoid T be a cancellative monoid in Theorem 2.3, then we can immediately get the translational hull of a completely 7*-simple semigroup which is the main theorem in [1]. Corollary 2.5 Let S = M[T;I, A; P| with P normalized, each entry in P is a unit of cancella- tive monoid T,and let e be the identity of T. Define a mapping o by a: (Ap) > (Ft,®) (A,p) € Q(5)) where F,t and ® are defined by the requirements (i, e, 1) = (Fa,--+ +) ES, (1) (le lp=(--,t,---) ES, (2) (le,wp=(-+,+++ we) € S. (3) Further define a mapping T by 7: (F,t,®) > (A,p) ((F,t,®) € T(S)), where and p are defined by the formulas Mi,h, w) = (Fi,tpiesh,w) ((t,h,m) € 8), (4) (i,h, w)p a (i, hpy, rit, u®) (i, h, 1) € S). (5) Then o and tT are mutually inverse isomorphisms between Q(S) and T(S). Also, if we let T be a group G in Theorem 2.3, then we can immediately get the translational hull of a completely simple semigroup which is the Theorem III.7.2 in [15]. On Translational Hull Of Completely 7*’~-Simple Semigroups 63 Corollary 2.6 Let S = M[G;I, A; P| with P normalized, and let e be the identity of group G. Define a mapping a by o:(A,p) > (Fig,®) (A, p) € Q(5)) where F,g and ® are defined by the requirements A(t, e, 1) = (Fi,--- +++) (1) (Le; lps (es Ges+*) (2) (le, uw) = (++ +++, w®) (3) Further define a mapping T by 7: (F,g,®) > (Ap) ((F.g,®) € T(S)), where and p are defined by the formulas Mi,h, hw) = (Fi,gpieih, pu) (i,h,m) € 8), (4) (i,h,u)p = (i, hpp,rig,uw®), (i,h,u) €S). (5) Then o and rt are mutually inverse isomorphisms between Q(S) and T(S). References 1] CHEN Y.Z., LI Y.H., The translational hull of completely 7*— simple semigroups, Journal of South China Normal University, 2007,1:28-31. 2] CLIFFORD A.H., PETRICH M., Some classes of completely regular semigroups, J. Alge- bra, 1977, 46:462—480. 3] FOUNTAIN J.B., Abundant semigroups, Proc London Math. Soc., 1982,44(3):103-129. 4) FOUNTAIN J.B., LAWSON M V., The translational hull of an adequte semigroup, Semi- group Forum, 1985, 32:79-86. 5] HOWIE J.M., Fundamentals of Semigroup Theory, Oxford:Clarendon Press,1995. 6] GONG C., ZHANG D., YUAN Y., The structure of completely 7**~— simple semigroups, Journal of Southwest Normal University(Natural Science Edition), 2011, 36(1): 21-25. 7| GUO X.J., SHUM K.P., On translational hulls of type A semigroups, J. Algebra, 2003, 269: 240-249. 8] GUO X.J., GUO Y.Q., The translational hull of a strongly right type A semigroup, Science in China, Ser A, Math, 2000, 43(1): 6-12. 9] GUO Y.Q., GONG C.M., REN X.M., A survey on the origin and developments of green’s relations on semigroups, Journal of Shandong University(Natural Science Edition), 2010, 45(8): 1-18. [10] GUO Y.Q., SHUM K.P., GONG C.M., On (x, ~)-Green’s relations and ortho-lc-monoids, Communications in Algebra, 2010, 39(1),5-31. 64 11 12 13 14 15 16 Yizhi Chen, Siyan Li and Wei Chen MCALISTER D.B., One-to-one partial right translations of a right cancellative semigroup, J Algebra, 1976,43:231-251. PASTIJN F., A representation of a semigroup by a semigroup of matrices over a group with zero, Semigroup Forum, 1975,10: 238-249. PETRICH M., The structure of completely regular semigroups, Trans Amer Math Soc., 1974, 189 : 211-236. PETRICH M., The translational hull in semigroups and rings, Semigroup Forum, 1970,1: 283-360. PETRICH M., REILLY N., Completely Regular Semigroups, Wiley, 1999. REN X.M., SHUM K.P., The structure of superabundant semigroups, Science in China, Ser A, Math, 2004. 47(5): 756-771. International J.Math. Combin. Vol.1(2015), 65-78 Some Minimal (r,2,/)-Regular Graphs Containing a Given Graph and its Complement N.R.Santhi Maheswari (Department of Mathematics, G.Venkataswamy Naidu College, Kovilpatti, India) C.Sekar (Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, India) E-mail: nrsmaths@yahoo.com, sekar.acas@gmail.com Abstract: A graph G is called (r,2, k)-regular graph if each vertex of G is at a distance 1 away from r vertices of G and each vertex of G is at a distance 2 away from k vertices of G [9]. This paper suggest a method to construct a ((m+2(n—1)), 2, (m—1)(2n—1)))-regular graph HA4 of smallest order 2mn containing a given graph G of order n > 2, and its complement G° as induced subgraphs, for any m > 1. Also, in this paper we calculate the topological indices Wiener index W, hyper Wiener index WW, degree distance DD, variance of degrees, first, second and third Zagreb indexes of the graphs H4 which we constructed in this paper. Key Words: Induced subgraph; clique number; independent number; (d, k)-regular graphs; (2, k)-regular graphs;(r, 2, k)-regular graphs; semiregular. AMS(2010): 05C12 §1. Introduction In this paper, we consider only finite, simple, connected graphs. For basic definitions and terminologies we refer Harary [7] and J.A.Bondy and U.S.R.Murty [4]. We denote the vertex set and edge set of a graph G by V(G) and E(G) respectively. The degree of a vertex v is the number of edges incident at v. A graph G is regular if all its vertices have the same degree. For a connected graph G, the distance d(u, v) between two vertices u and v is the length of a shortest (u,v) path. Therefore, the degree of a vertex v is the number of vertices at a distance 1 from v, and it is denoted by d(v). This observation suggests a generalization of degree. That is, da(v) is defined as the number of vertices at a distance d from v. Hence di(v) = d(v) and Na(v) denote the set of all vertices that are at a distance d away from v in a graph G. That is, Ni(v) = N(v) and No(v) denotes the set of all vertices that are at a distance 2 away from v in a graph G and closed neighbourhood Nv] = N(v) U {v}. The concept of distance d-regular graph was introduced and studied by G.S. Bloom, J.K. Kennedy and L.V.Quintas [3]. A graph G is said to be distance d-regular if every vertex of G has the same number of vertices at a distance d from it. A graph G is said to be (d, k)-regular 1Received August 12, 2014, Accepted February 22, 2015. 66 N.R.Santhi Maheswari and C.Sekar graph if da(v) = k, for all v in V(G). A graph G is (2, &) regular if dg(v) = k, for all v in V(G). The concept of the semiregular graph was introduced and studied by Alison Northup [2]. We observe that (2,k) - regular graph and & - semiregular graph are the same. A graph G is said to be (r, 2, k)-regular if d(v) =r and d2(v) =k, for all v € V(G). An induced subgraph of G is a subgraph H of G such that E(H) consists of all edges of G whose end points belong to V(H).In 1936, Konig [8] proved that if G is any graph, whose largest degree is r, then there is an r-regular graph H containing G as an induced subgraph. In 1963, Paul Erdos and Paul Kelly [6] determined the smallest number of new vertices which must be added to a given graph G to obtain such a graph. We now suggest a method that may be considered an analogue to Konig’s theorem for (r,2,)-regular graph. With this motivation, already we have constructed a (m+n -— 2), 2,(m—1)(n—1))-regular graph S' of order mn containing a given graph G of order n > 2 as an induced subgraph, for any m > 1 [12]. In this paper, our main objective is to construct a ((m+2(n—1)), 2, (m—1)(2n—1))- regular graph of smallest order 2mn containing the given graph G of order n > 2, and its complement G‘° as induced subgraphs, for any m > 1. §2. (r,2,k)-Regular Graph Definition 2.1 A graph G is called (r,2,k)-regular if each vertex in graph G is at a distance one from exactly r-vertices and at a distance two from exactly k vertices. That is, d(v) =r and dg(v) =k, for allv in G. Example 2.2 A few (r,2,k)-regular graphs are listed following. (1) The Peterson graph is a (3, 2, 6)-regular graph . (2) A complete bipartite graph Ky» is a (n,2,(n—1))-regular graph. Observation 2.3 For any n > 1, the smallest order of (n, 2, (n —1))- regular graph containing the complete bipartite graph K,,,, of order 2n is Ky», itself. The following facts can be verified easily. Observation 2.4((9]) If G is (r,2, k)-regular graph, then 0 << k < r(r—1). Observation 2.5((10]) For any r > 1, a graph G is (r, 2, r(r — 1))-regular if G is r-regular with girth at least five. Observation 2.6([11]) For any odd r > 3, there is no (r,2, 1)-regular graph. Observation 2.7([{11]) Any (r,2,%)-regular graph has at least k + r+ 1 vertices. Observation 2.8({11]) If r and k are odd, then (r,2,k)-regular graph has at least k + r +2 vertices. Observation 2.9([12]) For any m > 1, every graph G of order n > 2 is an induced subgraph of (n +m — 1,2, (mn — 1))-regular graph H of order 2mn. Observation 2.10({13]) For any m > 1, every graph G of order n > 2 is an induced subgraph of (n+ m — 2,2, (m—1)(n — 1))-regular graph H of order mn. Some Minimal (r, 2, k)-Regular Graphs Containing a Given Graph and its Complement 67 §3. Minimal (r,2,)-Regular Graphs Containing Given Graph and Its Complement as an Induced Subgraph In this section we construct a smallest (r,2,k)-regular graphs containing given graph and its complement as an induced subgraph. Theorem 3.1 For a graph G of order n > 2, there exists a (m+ 2(n — 1), 2, (m-— 1)(2n — 1))- * 9 regular graph H4 of order 2mn such that G and G° are the induced subgraphs of H4. Proof Let G be a graph of order n > 2, G and G° has the same vertex set {uv} :1 <i < n}. Take a graph G’ which is isomorphic to G°. The vertex set of G’ is denoted as {uj : 1 < i < n} and u} corresponds to v}(1 < i < n). Let G. = GUG?". Then V(G1) = {v},u} : 1 <i <n}. Let Gi(2 < t < m) be the (m — 1) copies of Gi with the vertex set V(G;) = jou: PS oe ph, for (2< 6 <-m)-and vi, ui(2 < f < mm) correspond to vi,us(1 <4 < a? a n) respectively. The desired graph Hy has the vertex set V(H4) = U V(G;), and edge set t=1 E(H4) = U E(G:) U Fy U E> U E3 U E4 U Es, where, t=1 Ey ai Bs U “foto it) omy} : vty} ¢ E(Gi)(1<j<n),G@+1<i<n)}, Ex = U {vivt? :(1<i<m-1),(1<j<m-d}, 1 > Il 1 er {uiust?, uu} : ulud ¢ B(Gi)1 <j <n),(G+1<i<n)}, 3 ] Ufujup (0 <i<m—1),(<j7<m—i}, & | ice Es = U “Yutu pha 1 Sag s nh: The resulting graph H4 contains G; as an induced subgraph. More over in Hy,(1 < t < m), d(uf) = m+ 2(n—1), for (1 <i <n). Then Hy is m+ 2(n — 1), regular graph with 2mn vertices. Hence H contains G and G® as induced subgraphs. In H4, d(v;) = d(vj) = d(ui) d(u}) =m+2n—2,1<i<n. To find the dz degree of each vertex in Hy, the following cases are examined. Case 1. t=1.Ifu Ee V(G;), then v € V(G) (or) v € V(G’). Subcase 1.1 If v € V(G), then v = vj, for some j. Let vj € V(H4) — N[vu;]. Then vj and v; are non-adjacent vertices in H4. By our construction, v; is adjacent to v7 and vj is adjacent to vy. Then d(vj,v;) = 2. Hence vj € No(v; v) This implies that Va N[v}] C No(v}). If v; € No(v;), then vj is non- si sioatk with v;. This implies that v; ¢ V(H1) — N[v;]. Hence Naot) = V(Ha) — Nie], 1 <4<n) and da(od) = (1) @n—1), (1 <4 < 0). Subcase 1.2 If v €¢ V(G’), then v = uj, for some j. Let uj € V(Ha) — N[u;]. Then, uj and uj are non-adjacent vertices in H4. By our con- struction, uj is adjacent to uj and uj is adjacent to u;. Then d(uj,u;) = 2. Hence uj; € No(u i): This irnplies that V(H4) — N[uj] C No(uj). if uj € No(uj), iio uj is non- adjacent with uj Hence u; € V(H4) — N[uj]. This implies that No(u 1) = V(M4) — Nful], (1 < i < n) and 68 N.R.Santhi Maheswari and C.Sekar do(uj) = (m—1)(2n—-1),(1<i<n). Case 2. 2<t<m-—1. Ifve V(G), then v = vj (or) v = uf, for some j. Subcase 2.1 If v = vj and if vi € V(H4) — N[v;], then vj and v; are non-adjacent vertices in H4. By our construction, v‘ is adjacent to vj and vj is adjacent to vj. Then d(vj,v;) = 2. Hence vj € No(v}). This implies that V(H4) — N[vj] No(v;). If vf € No(v}), then, vj is non-adjacent with v;. This implies that vf ¢ V(H1) — N[v;]. Hence No(v;) = V(H4) — N[vj], (1 <i <n) and do(v}) = (m—1)(2n—1),(1<i<n). Subcase 2.2 Ifv =u‘ and if ui ¢ V(H4)—N[uj], then u§ and u; are non-adjacent vertices in H4. By our construction, u‘ is adjacent to uf? and u;*" is adjacent to uj. Then d(u‘, uj) = 2. Hence ui € N2(u;). This implies that V(H4) — N[u;] © No(u;). If uf © No(uj), then, uj is non-adjacent with u;. Hence uf €¢ V(H4) — N[{u;]. This implies that No(u;) = V(H4) — N[uj], (1 <i <n) and do(u}) = (m—1)(2n—-1), (1 <i<n). Case 3. t=m. Ifv Ee V(Gm), then v = vj" (or) v = ul" for some j. Subcase 3.1 If v = v7" and if uv" € V(H4)— N[v;], then uv)” and v; are non-adjacent vertices in H4. By our construction, v4” is adjacent to v}” and v}" is adjacent to vj. Then d(v}", vu; ) = 2. Hence v3" € No(v;). This implies that V(H4) — N[v;] C No(v;). If v7" € No(v;), then v3” is non-adjacent with v}. Hence v7” € V(H4) — N[v}]. This implies that No(vj) = V(H4) — N[v}), 1<i<nand do(v}) = (m—1)(Q2n—-1),(1<i<n). Subcase 3.2 If v = u'" and if uv” € V(A1)—N{uj], then wu” and u; are non-adjacent vertices in Hq. By our construction, u’” is adjacent to uj” and uj” is adjacent to u;. Then d(u%”, uj) = 2. Hence u%” € No(v;). This implies that V(H4) — N[ui] C No(us). If uw” € No(u;), then u%” is non-adjacent with u;. Hence wu” € V(H4) — N[u;]. This implies that No(u;) = V(Ha) — N[uj], 1 <i<nand do(u}) = (m—1)(2n—1), (1 <i < n). Similarly for (1 < t < m)d2(v') = do(ut) = (m—1)(2n—1),(1 <i <n). Hy is (m+ 2(n — 1), 2, (m — 1)(2n — 1))-regular graph of order 2mn containing a given graph G of order n > 2 and its complement as induced subgraphs. Corollary 3.2 For any m > 1, the smallest order of (m+ 2(n— 1), 2, (m— 1)(2n — 1))-regular graph containing a given graph of order n > 2 and its complement is 2mn. Proof For the graph H4 constructed in Theorem 3.1 is (m+ 2(n — 1), 2, (m— 1)(2n — 1))- regular graph of order 2mn. Suppose H4 is (m+ 2(n — 1), 2, (m — 1)(2n — 1))-regular graph of order 2mn —1. Then, for each v; € Hy, do(vi) = (m — 1)(2n — 1) and d(u;) = m + 2(n — 1). Hence Hy, has at least ((m — 1)(2n— 1) + (m+ 2(n—1)+1) = 2mn vertices, a contradiction. Figure 1 Corollary 3.3 Every graph G of order n > 2, and its complement G° are the induced sub-graphs Some Minimal (r,2,k)-Regular Graphs Containing a Given Graph and its Complement 69 of (2n, 2,(2n — 1))-regular graph of smallest order 4n. In Figure 1, Corollary 3.3 is illustrated for G = Ks, in which the graph G is induced by the vertices x, y, Z. Corollary 3.4 Every graph G of order n > 2, and its complement G° are the induced subgraphs of (2n + 1,2,2(2n — 1))-regular graph of smallest order 6n. | i ) LK] 4 ) p \} W yf iy G- G=K; Go = Ks Mh a A 17 2 (on KY N ETH Y M Hh Ay Figure 2 In Figure 2, Corollary 3.4 is illustrated for G = Kg and G = Ks, in which the graph G and Gis induced by the vertices 7, y for G = K2. In the second graph, the graph G and G‘is induced by the vertices x,y, z for G = K3. Corollary 3.5 Every graph G of order n > 2, and its complement G° are the induced subgraphs of (2n + 2,2,3(2n —1))-regular graph of smallest order 8n. Corollary 3.6 Every graph G of order n > 2, and its complement G° are the induced subgraph of (2n + 3, 2,4(2n — 1))-regular graph of smallest order 10n. Remark 3.7 There are at least as many (m+2(n— 1), 2, (m—1)(2n—1))-regular of order 2mn as there are graphs G of order n > 2. If m = 2,3,4,5,..., then there are (2n, 2, (2n—1)), (2n+ 1,2, 2(2n — 1)), (Qn + 2, 2,3(2n — 1)), (2n + 8, 2, 4(2n — 1)),... regular graphs of smallest order 4n, 6n, 8n, 10n,12n... respectively containing any graph G of order n > 2 and its complement as induced subgraphs. §4. Topological Indices of the Graph H, The topological indices Wiener Index W, Hyper Wiener Index WW, Degree Distance DD, Variance of degrees, The first Zagreb index, The second Zagreb Index and the third Zagreb Index of the graph H4, which was constructed in Theorem 3.1 are calculated in this section. Topological index Top(G) of a graph G is a number with this property that for every graph HT isomorphic to G, Top(G) = Top(H). For historical background, computational techniques and mathematical properties of Zagreb indices and Wiener, Hyper Wiener one can refer to [21,22,23,24,25]. The graph H4 is (m+ 2n — 2,2, (m— 1)(2n — 1))-regular graph having 2mn vertices and mn(m + 2n — 2)) edges with diameter 2. Also, for each v € Hy, do(v) = (m— 1)(2n — 1) and 70 N.R.Santhi Maheswari and C.Sekar d(v) =m+ 2n-—- 2. Computation of W, WW and DD for Hy, is done by using the following theorem [14]: Let G be a graph with n vertices, m edges and with diameter 2, then (1) W(G) = n(n —1)—m; (2) WW(G) = 3/2(n(n — 1)) — 2m; (3) DD(G) = 4(n — 1)m — M,(G). The Wiener index W is the first and important topological index in chemistry which was introduced by H. Wiener in 1947 to study the boiling points of parafins. This index is useful to describe molecular structures and also crystal lattice that depends on its W value. Definition 4.1 The Wiener index, W(G) of a finite, connected graph G is defined to be 1 W(G) = 5 Sy d(u,v), where d(u,v) denotes the distance between u and v in G. Wiener Index of a graph Hy = W/(H4) = 2mn(2mn-— 1) — ((mn)(m + 2(n — 1)) mn(4mn — 2 —m — 2n + 2) = (mn)(4mn — (m + 2n)) l| The Hyper Wiener index WW was introduced by Randic. The Hyper Wiener Index WW is used as a structure descriptor for predicting physicochemical properties of organic compounds. Definition 4.2 The Hyper Wiener index WW(G) of a finite, connected graph G is defined to 1 be WW(G) = 3 (WilG) + W2(G)), where Wi(G) = W(G) and Wy(G) = >> da(k)(k*) is called the Wiener-type invariant of G associated to a real number. I Hyper Wiener Index of a graph H4 WW (H4) (3/2)(2mn(2mn — 1) — 2mn(m + 2(n — 1))) = (mn)(6mn —3— 2m —4n+4) (mn)( mn)(6mn — (2m + 4n) + 1) The Zagreb indices were introduced by Gutman and Trinajestic [7,10,14]. Definition 4.3 The oldest and most investigated topological graph indices are defined as: First Zagreb index Mi(G) = di vev(a) (dg(v))?, second Zagreb index Mz(G) = we R(a) (de(u)de(v)) and third Zagreb index M3(G) = >> |d(w) — d(v)|, uv € E(G). Some Minimal (r, 2, k)-Regular Graphs Containing a Given Graph and its Complement 71 The Zagreb Indices of graph H4 are 1. Mi(H4) = S-d(u)d(u) = S°d(u)? = 2mn((m + 2n — 2)?) 2. Mo(Hs) = 5 d(u)d(v), uv € E(As) n) mn) = (mn)(m ae 1)(m + 2(n — 1)(m + 2(n— 1) = ((m + 2(n — 1))°) 3. M3(H4) = S~|d(u) —d(v)||, uv € E(Ha) = So \(m+2(n-1))—(m+2(n-1))| =0. = Definition 4.4((4]) The degree distance (Schultz index) of G was introduced by Dobrynin and Kochetova and Gutman as a weighted version of the Wiener index defined as DD(G) = Yo (d(u) + d(v))d(u, v). It is to be noted that DD(G) and W(G) are closely mutually related for certain classes of molecular graphs. The degree distance of graph Hy is DD(H4) = 4(2mn— 1)(mn)(m + 2(n — 1)) — Mi (Ag) = 2mn(m +t 2n — 2)[2(2mn — 1) — (m+ 2n — 2)) = 2mn(m-+ 2n — 2)[4mn — (m+ 2n)] Definition 4.5([13]) The status, or distance sum of a vertex v in a graph is defined by s(v) = J d(u,v), where d(u,v) is the distance between the vertices u andv andu#v. The status sequence of a graph consists of a list of the stati of all the vertices. Since diameter of Hy, is two, the status of a vertex v in H, is s(v) I (m + 2(n — 1) + 2(m — 1)(2n — 1) m+ 2n—2+4+4mn— 2m — 4n4+ 2 = 4mn — 2(m +n) I Definition 4.6 A graph is said to be self-median, or SM, if the stati of its vertices are all equal. Every vertex in H4 has the same status 4mn — 2(m-+n). Whence, Hy is a self-median graph. §5. Open Problems For further investigation, the following open problem is suggested: (1) Construct (r,m,k)-regular graphs containing a given graph G and its complement of order n > 2, as induced subgraph, for m > 3. (2) Construct (r,m,k)-regular graphs containing a given graph G and its complement of order n > 2, as induced subgraph, for all values of k. 72 N.R.Santhi Maheswari and C.Sekar References 1] Yousef Alavi, Gary Chartrand, F.R.K. Chang, Paul Erdos, H.L.Graham and O.R. Oeller- mann, J. Graph Theory, 11(2) (1987), 235-249. 2} Alison Northup, A Study of Semiregular Graphs, Bachelor Thesis, Stetson University (2002). 3] G.S. Bloom, J.K. 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Comput. Chem., 54 (2005) 233-239. International J.Math. Combin. Vol.1(2015), 74-79 On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs P. Siva Kota Reddy!, P. N. Samanta? and Kavita S Permi® 1. Department of Mathematics, Siddaganga Institute of Technology, B.H.Road, Tumkur-572 103, India 2. Department of Mathematics, Berhampur University, Berhampur-760 007, Orissa, India 3. Department of Mathematics, Acharya Institute of Technology, Soladevanahalli, Bangalore-560 107, India E-mail: reddy.math@yahoo.com; pskreddyQsit.ac.in Abstract: In this paper, we obtained a characterization of signed graphs whose jump signed graphs are switching equivalent to their two path signed graphs. Key Words: Smarandachely k-signed graph, t-Path graphs, jump graphs, signed graphs, balance, switching, t-path signed graphs and jump signed graphs. AMS(2010): 05C38, 05022 §1. Introduction For standard terminology and notation in graph theory we refer Harary [6] and Zaslavsky [20] for signed graphs. Throughout the text, we consider finite, undirected graph with no loops or multiple edges. Given a graph I and a positive integer t, the t-path graph (T), of I is formed by taking a copy of the vertex set V(T) of , joining two vertices u and v in the copy by a single edge e = wv whenever there is a u— v path of length t in T. The notion of t-path graphs was introduced by Escalante et al. [4]. A graph G for which (Dy re (1) has been termed as t-path invariant graph by Esclante et al. in [4], Escalante & Montejano [5] where the explicit solution to (1) has been determined for t = 2,3. The structure of t-path invariant graphs are still remains uninvestigated in literature for all t > 4. The line graph L(T) of a graph T = (V,£) is that graph whose vertices can be put in one-to-one correspondence with the edges of I’ so that two vertices of L(I’) are adjacent if, and only if, the corresponding edges of T° are adjacent. The jump graph J(L) of a graph T = (V,E) is L(L), the complement of the line graph L(T) of T (see [6]). A Smarandachely k-signed graph is an ordered pair S = (G,o) (S = (G,p)) where G = (V,F) is a graph called underlying graph of S and o : E — (@1,@2,--: ,@n) (u: V 1Received June 9, 2014, Accepted February 26, 2015. On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs 75 (€1,€2,:-+ ,@)) is a function, where each @; € {+,—}. Particularly, a Smarandachely 2-signed graph is called abbreviated to signed graph, where T = (V,£) is a graph called underlying graph of % ando : E — {+,—} is a function. We say that a signed graph is connected if its underlying graph is connected. A signed graph © = (T,¢) is balanced, if every cycle in © has an even number of negative edges (See [7]). Equivalently, a signed graph is balanced if product of signs of the edges on every cycle of © is positive. Signed graphs ©; and No are isomorphic, written ©, = No, if there is an isomorphism between their underlying graphs that preserves the signs of edges. The theory of balance goes back to Heider [8] who asserted that a social system is balanced if there is no tension and that unbalanced social structures exhibit a tension resulting in a tendency to change in the direction of balance. Since this first work of Heider, the notion of balance has been extensively studied by many mathematicians and psychologists. In 1956, Cartwright and Harary [3] provided a mathematical model for balance through graphs. For more new notions on signed graphs refer the papers (see [12-17, 20]). A marking of X is a function ¢: V([T) — {+,—}. Given a signed graph © one can easily define a marking ¢ of © as follows: For any vertex v € V(X), uve E(X) the marking ¢ of & is called canonical marking of U. A switching function for © is a function ¢ : V — {+,—}. The switched signature is aS(e) := ¢(v)o(e)¢(w), where e has end points v, w. The switched signed graph is ©$ := (S|). We say that © switched by ¢. Note that US$ = U~S (see [1]). If X CV , switching © by X (or simply switching X) means reversing the sign of every edge in the cut set E(X, X°). The switched signed graph is ©*. This is the same as ©$ where ¢(v) := — if and only if v € X. Switching by ¢ or X is the same operation with different notation. Note that 5* = 5*°*, Signed graphs ©; and Nz are switching equivalent, written 1 ~ Xz if they have the same underlying graph and there exists a switching function ¢ such that y¢ 2 yy. The equivalence class of &, [SE] = {b': Dw OF, is called the its switching class. Similarly, 4; and “2 are switching isomorphic, written 41 & No, if 44 is isomorphic to a switching of Nz. The equivalence class of © is called its switching isomorphism class. Two signed graphs ©; = ((1,01) and Nz = (T2, 02) are said to be weakly isomorphic (see [18]) or cycle isomorphic (see [19]) if there exists an isomorphism ¢ : Ty — T2 such that the sign of every cycle Z in 41 equals to the sign of 6(Z) in Sg. The following result is well known (see [19]). Theorem 1.(T. Zaslavsky, [19]) Two signed graphs 1 and Ng with the same underlying graph are switching equivalent if and only if they are cycle isomorphic. 76 P. Siva Kota Reddy, P. N. Samanta and Kavita S Permi In [11], the authors introduced the switching and cycle isomorphism for signed digraphs. The notion of t-path graph of a given graph was extended to the class of signed graphs by Mishra [9] as follows: Given a signed graph & and a positive integer t, the t-path signed graph (=); of © is formed by taking a copy of the vertex set V(X) of 4, joining two vertices u and v in the copy by a single edge e = uv whenever there is au—v path of length t in S and then by defining its sign to be — whenever in every u—v path of length t in % all the edges are negative. In [13], P. S. K. Reddy introduced a variation of the concept of t-path signed graphs studied above. The motivation stems naturally from one’s mathematically inquisitiveness as to ask why not define the sign of an edge e = uv in (%); as the product of the signs of the vertices u and vin &. The t-path signed graph (X), = ((T)1,0’) of a signed graph © = (I,c) is a signed graph whose underlying graph is (I), called t-path graph and sign of any edge e = uv in (%)¢ is p(u)u(v), where yz is the canonical marking of 4. Further, a signed graph © = (T,¢) is called t-path signed graph, if % & (’),, for some signed graph \’. In this paper, we follow the notion of t-path signed graphs defined by P. S. K. Reddy as above. Theorem 2.(P. S. K. Reddy, [13]) For any signed graph % = (T,c), its t-path signed graph (X), ts balanced. Corollary 3. For any signed graph & = (To), its 2-path signed graph (%)2 is balanced. The jump signed graph of a signed graph S = (G,c) is a signed graph J(S) = (J(G),0’), where for any edge ee’ in J(S), a’(ee’) = a(e)o(e’). This concept was introduced by M. Acharya and D. Sinha [2] (See also E. Sampathkumar et al. [10]). Theorem 4.(M. Acharya and D. Sinha, [2]) For any signed graph % = (Ic), its jump signed graph J(X) is balanced. §2. Switching Equivalence of Two Path Signed Graphs and Jump Signed Graphs The main aim of this paper is to prove the following signed graph equation (X)2 ~ J(d). We first characterize graphs whose two path graphs are isomorphic to their jump graphs. Theorem 5. A graph T = (V,E) satisfies (T)2 = J(T) if, and only if, G = Cy or Cs. Proof Suppose T is a graph such that ([)2 = J(I). Hence number of vertices and number of edges of [ are equal and so [ must be unicyclic. Let C = Ci, be the cycle of length m > 3 inf. Case l. m=3. Let V(C) = {u1,u2,u3}. Then C is also a cycle in (I)2, where as in J(I), the vertices On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs 77 corresponds the edges of C are mutually non-adjacent. Since (()2 = J(T), J(T) must also contain a C3 and hence I must contain 3K, disjoint union of 3 copies of Kg. Whence I’ must contain either [1,T2 or 3 as shown in Figure 1 as induced subgraph. ry Tr rs Figure 1. Subcase 1.1 If I contains [2 or 3. Let v be the vertex satisfying d(v,C) = 2. Since d(v,C) = 2 there exists a vertex u in T adjacent to v and a vertex in C' say uy. Now, in (T)a, the vertex v is not adjacent to u. Since C is also a cycle in (I)z and w is adjacent to uy in IT, the vertices w ug and u3 forms a cycle C’ in ([')g. Further, the vertex v is not adjacent to C’. Hence, I contains H = C3 U K, as a induced subgraph. But since IY = K 1,3 Which is a forbidden induced subgraph for line graph L([) and J(T) = L(T), we must have ([)2 % J(T). Subcase 1.2 If T contains [;. Then by subcase(i), f =I. But clearly, (T)2 4 J(T). Case 2. m> 4. Suppose that m > 4 and there exists vertex v in T which is not on the cycle C. Let C = (v1, V2, U3, V4, ---) Um; U1). Since T is connected v is adjacent to a vertex say, v; in C. Then the subgraph induced by the vertices v;-1,v,U;41 and vu; in ([)2 is K3 U Ky. Now the graph K30Kq is K1,3 which is a forbidden induced subgraph for L(T) = J(P). Hence I is not a jump graph. Hence [ must be a cycle. Clearly ([)2(C4) = 2K2 = J(C4) and J(C5) = (T)2(Cs) = Cs it remains to show that for T = C,,, with m > 6 does not satisfy J(Cm) = (T)2(Cm), for m > 6. Suppose that m > 6. Then clearly every vertex in J(C,,) is adjacent to at least m—2 > 4 vertices where as in ([')2, degree of every vertex is 2. This proves the necessary part. The converse part is obvious. We now give a characterization of signed graphs whose two path signed graphs are switching equivalent to their jump signed graphs. Theorem 6. For any signed graph © = (T,0), (S)2 ~ J(5) if, and only if, the underlying graph TV is either Cs or C4. Proof Suppose that (X)z ~ J(X). Then clearly, (T)2 = J(L). Hence by Theorem 5, [ must be either Cy or Cs. Conversely, suppose that © is a signed graph on C4 or Cs. Then by Theorem 5, (()2 & J(T). 78 P. Siva Kota Reddy, P. N. Samanta and Kavita S Permi Since for any signed graph %, by Corollary 3 and Theorem 4, (©)2 and J() are balanced, the result follows by Theorem 1. Remark The only possible cases for which ())2 ~ J(X) are shown in Figure 2. Figure 2. References 1] R. P. Abelson and M. J. Rosenberg, Symoblic psychologic:A model of attitudinal cognition, Behav. Sci., 3 (1958), 1-13. 2] M. Acharya and D.Sinha, A characterization of signed graphs that are switching equivalent to other jump signed graphs, Graph Theory Notes of New York, XLIII(1) (2002), 7-8. 3] D. Cartwright and F. Harary, Structural Balance: A Generalization of Heider s Theory, Psychological Review, 63 (1956), 277-293. 4] F. Escalante, L. Montejano and T. Rojno, A characterization of n-path connected graphs and of graphs having n-th root, J. Combin. Th., Ser. B, 16(3) (1974), 282-289. 5] F. Escalante and L. Montejano, Trees and n-path invariant graphs, Graph Theory Newslet- ter, 3(3), 1974. 6] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969. 7| F. Harary, On the notion of balance of a signed graph, Michigan Math. J., 2 (1953), 143-146. 8] F. Heider, Attitudes and Cognitive Organisation, Journal of Psychology, 21 (1946), 107- 112. 9] V. Mishra, Graphs associated with (0,1) and (0,1, —1) matrices, Ph.D. Thesis, IIT Bombay, 1974. [10] E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya, Jump symmetric n-signed graph, Proceedings of the Jangjeon Math. Soc., 11(1) (2008), 89-95. 11 12 13 14 15 16 17 18 19 20 On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs 79 E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy, Characterization of line sidigraphs, Southeast Asian Bull. Math., 35(2) (2011), 297-304. P. Siva Kota Reddy, S. Vijay and V. Lokesha, n*” Power signed graphs, Proceedings of the Jangjeon Math. Soc., 12(3) (2009), 307-313. P. Siva Kota Reddy, t-Path Sigraphs, Tamsui Ozford J. of Math. Sciences, 26(4) (2010), 433-441. P. Siva Kota Reddy, B. Prashanth, and T. R. Vasanth Kumar, Antipodal Signed Directed Graphs, Adun. Stud. Contemp. Math., 21(4) (2011), 355-360. P. Siva Kota Reddy and B. Prashanth, S-Antipodal Signed Graphs, Tamsui Oxf. J. Inf. Math. Sci., 28(2) (2012), 165-174. P. Siva Kota Reddy and U. K. Misra, The Equitable Associate Signed Graphs, Bull. Int. Math. Virtual Inst., 3(1) (2013), 15-20. P. Siva Kota Reddy and U. K. Misra, Graphoidal Signed Graphs, Adun. Stud. Contemp. Math., 23(3) (2013), 451-460. T. Sozansky, Enueration of weak isomorphism classes of signed graphs, J. Graph Theory, 4(2)(1980), 127-144. T. Zaslavsky, Signed graphs, Discrete Appl. Math., 4(1) (1982), 47-74. T. Zaslavsky, A mathematical bibliography of signed and gain graphs and its allied areas, Electronic J. Combin., 8(1) (1998), Dynamic Surveys (1999), No. DS8. International J.Math. Combin. Vol.1(2015), 80-85 A Note on Prime and Sequential Labelings of Finite Graphs Mathew Varkey T.K (Department of Mathematics, T.K.M. College of Engineering, Kollam-5, India) Sunoj. B.S (Department of Mathematics, Government Polytechnic College, Ezhukone, Kollam, Kerala, India) E-mail: mathewvarkeytk@gmail.com, spalazhi@yahoo.com Abstract: A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy a label depending on the vertex labels f(x) and f(y). In this paper, we study some classes of graphs and their corresponding labelings. Key Words: Labeling, sequential graph, harmonious graph, prime graph, Smarandache common k-factor labeling. AMS(2010): 05C78 §1. Introduction Unless mentioned or otherwise, a graph in this paper shall mean a simple finite graph without isolated vertices. For all terminology and notations in Graph Theory, we follow [5] and all terminology regarding to sequential labeling, we follow [3]. Graph labelings where the vertices are assigned values subject to certain conditions have been motivated by practical problems. Labeled graphs serves as useful mathematical models for a broad range of applications such as coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolutional codes with optimal autoconvolutional properties. They facilitate the optimal nonstandard encodings of integers. Labeled graphs have also been applied in determining ambiguities in X-ray crystallographic analysis, to design a communication network addressing system, in determining optimal circuit layouts and radio astronomy problems etc. A systematic presentation of diverse applications of graph labelings is presented in [1]. Let G be a (p,q)-graph. Let V(G), E(G) denote respectively the vertex set and edge set of G. Consider an injective function g : V(G) — X, where X = {0,1,2,---,q} if G is a tree and X = {0,1,2,---,q—1} otherwise. Define the function f* : E(G) — N, the set of natural numbers such that f*(uv) = f(u) + f(v) for all edges uv. If f*(E(G)) is a sequence of distinct consecutive integers, say {k,k+1,---,k+q—1} for some k, then the function f is said to be sequential labeling and the graph which admits such a labeling is called a sequential graph. Another labeling has been suggested by Graham and S Loane [4] named as harmonious 1Received January 21, 2014, Accepted February 27, 2015. A Note on Prime and Sequential Labelings of Finite Graphs 81 labeling which is a function h : V(G) — Zq, q is the number of edges of G such that the induced edge labeling given by g*(uv) = [g(w) + g(v)] (mod gq) for any edge wv is injective. The notion of prime labeling of graphs, was defined in [6]. A graph G with n-vertices is said to have a prime labeling if its vertices are labeled with distinct integers 1, 2,--- ,n such that for each edge uv the labels assigned to u and v are relatively prime. Such a graph admitting a prime labeling is known as a prime graph. Generally, a Smarandache common k-factor labeling is such a labeling with distinct integers 1,2,---,n such that the greatest common factor of labels assigned to u and v is k for Vuu € E(G). Clearly, a prime labeling is nothing else but a Smarandache common 1-factor labeling. A graph admitting a Smarandache common k-factor labeling is called a Smarandache common k-factor graph. Particularly, a graph admitting a prime labeling is known as a prime graph in references. Notation 1.1 (a,b) =1 means that a and b are relatively prime. §2. Cycle Related Graphs In [2], showed that every cycle with a chord is graceful. In [9] proved that a cycle C,, with a chord joining two vertices at a distance 3 is sequential for all odd n,n > 5. Now, we have the following theorems. Theorem 2.1 Every cycle C,,, with a chord is prime, for all n > 4. Proof Let G be a graph such that G = C, with a chord joining two non- adjacent vertices of C,, for all n greater than or equal to 4. Let {v1,v2,---,un} be the vertex set of G. Let the number of vertices of G be n and the edges be n+ 1. Define a function f : V(G) > {1,2,---n} such that f(v;) = i,i = 1,2,---,n. It is obvious that (f(u), f(vigi)) = 1 for all ¢=1,2,---,(n—1). Also (1,n) = 1 for all n greater than 1. Now select the vertex v1 and join this to any vertex of C,, which is not adjacent to v;,G admits a prime labeling. -1 Theorem 2.2 Every cycle Cy, with | —1 chords from a vertex is prime, for all n greater than or equal to 5. Proof Let G bea graph such that G = C,,, n greater than or equal to 5. Let {v1, v2,..., Un} be the vertex set of G. Label the vertices of C;, as in Theorem 2.1. Next select the vertex v2. By our labeling f(v2) = 2. Now join v2 to all the vertices of C,, whose f-values are odd. Then it is clear that there exists exactly [25+] — 1 chords, and G admits a prime labeling. 10 4 7 6 Figure 1 82 Mathew Varkey T.K and Sunoj. B.S Remark 2.1 From Theorem 2.1, it is clear that there is possible to get n — 3 chords and Theorem 2.2 tells us there are [45+] — 1 chords. Thus the bound n — 3 is best possible and all other possible chords of less than these two bounds. Example 2.1 Figure 1 gives the prime labeling of C1 with 4-chords. Theorem 2.3 The graph Cy, + Kis is sequential for all odd n,n > 3. Proof Let v1, v2,--+ ,Un (n is odd) be the set of vertices of C,, and u, ui, Ua,--: , uz be the t+ 1 isolated vertices of Kit. Let G=Cy, + Kis and note that, G has n+¢t+ 1 vertices and n(2+t) edges. Define a function f : V(G) — {0,1,2,--- , 45+ + tn} such that a f(va-1) = i-1, for i=1,2,---, 75 n—-1 . ; n—1 f (vai) = 5 +4, fori =1,2,---, 5 3 f(u1) = rie -—1 and f(ui) = uae dry Go OB ae 34 We can easily observe that the above defined f is injective. Hence f becomes a sequential labeling of C;, + Kis. Thus C,, + Kit is sequential for all odd n, n > 3. Corollary 2.4 The graph Cy, + Kis is harmonious for all odd n > 3. Proof Any sequential is harmonius implies that C,, + K 1,4 is harmonius, n > 3. Theorem 2.5 The graph Cy, + Kias is sequential and harmonius for all odd n,n > 3. Theorem 2.6 The graph C, + Kiaat is sequential and harmonius for all odd n, n > 3. Example 2.2 Figure 2 gives the sequential labeling of the graph Cs + K1,1,13- Figure 2 A Note on Prime and Sequential Labelings of Finite Graphs 83 Theorem 2.7 The graph C, + Kia. is sequential as harmonius for odd n, n > 3. Theorem 2.8 The graph C), + Kiman is sequential and harmonius for all odd n, n > 3,m > 1. §3. On Join of Complete Graphs In [7], it is shown that L,, + Ky and B, + Ky are prime and join of any two connected graphs are not odd sequential. Now, we have the following. Theorem 3.1 The graph Ky, + Ke ts prime for n > 4. Proof Let G = Ki »+K2. We can notice that G has (n + 3)-vertices and (3n + 2)-edges. Let {w,v1,v2,+++ ,Un} be the vertices of Ky, and {ui,u2} be the two vertices of K2. Assign the first two largest primes less than or equal to n + 3 to the two vertices of Kz. Assign 1 to w and remaining n values to the n vertices arbitrarily, we can obtain a prime numbering of Kin + Ko. Corollary 3.1 The graph Ki + Ko is prime for all n > 4. §4. Product Related Graphs Definition 4.1 Let G and H be graphs with V(G) = Vi and V(H) = V2. The cartesian product of G and H is the graph GOH whose vertex set is Vi x V2 such that two vertices u = (x,y) and v = (a’,y’) are adjacent if and only if either x = x' and y is adjacent to y’ in H ory=y' and x is adjacent to x’ in G. That is, u adj v in GOH whenever [x = x’ and y adj y’] or [y = y' and x adj x'}. In [8] A-Nagarajan, A.Nellai Murugan and A.Subramanian proved that P,OK 2, P,OP, are near mean graphs. Definition 4.2 Let P, be a path on n vertices and K4 be a complete graph on 4 vertices. The Cartesian product P,, and K4 is denoted as P,OK4 with 4n vertices and 10n — 4 edges. Theorem 4.1 The graph P,OK4 is sequential, for alln > 1. Proof Let G = P,OK4. Let {vi 1, vi,2, 01,3, 01,4/1 = 1,2,--- ,n} be the vertex set of G. 84 Mathew Varkey T.K and Sunoj. B.S Define a function f : V(G) > {0,1,2,--- ,5n — 1} such that f(va-11) =10i-6 ; 1<i<s ifn isevenorl<i< 5 if n is odd. flix) =106—1) + 1<i<t ifn is even or <i M4 if n is odd. f(vai-1,3) = 101-9; 1<i<t if nis even or <i M4 if n is odd. flva-i1a) = 106-8 ; 1<i<t ifn is even or <i MSA if n is odd. f(voia) =10i-4 ; 1<i<g ifn is even or <i < if n is odd. f(ve2) = 106-1 ; 1<i<t if n is even or <i < " if n is odd. f(v2i3) =10i- 3 ; 1<i<t if nis even or <i < if n is odd. and mia 3. Te ee een eee Se nieed (a) Clearly we can see that f is injective. (b) Also, maxyey f(v) = max{max; 10i—6; max; 10(i—1); max; 10i—9; max; 10i—8; max; 10% —4; max; 10i—1; max; 10i—3; max; 10i—5} = 5n—1. Thus, f(v) = {0,1,2,...,5n—1}. Finally, it can be easily verified that the labels of the edge values are distinct positive integers in the interval [1,10n — 4]. Thus, f is a sequential numbering. Hence, the graph G is sequential. Example 4.1 Figure 4 gives the sequential labeling of the graph PyO ky. Figure 3 Corollary 4.1 The graph P,OK, is harmonius, for n > 2. A Note on Prime and Sequential Labelings of Finite Graphs 85 References 1] G.S.Bloom and S.W.Golomb, Pro. of the IEEE, 165(4), (1977) 562-70. 2] C.Delmore, M.Maheo. H.Thuiller, K.M.Koh and H.K.Teo, Cycles with a chord are graceful, Jour. of Graph Theory, 4 (1980) 409-415. 3] T.Grace, On sequential labeling of graphs, Jour. of Graph Theory, 7 (1983) 195-201. 4) R.L.Graham and N.J.A.Sloance, On additive bases and harmonious graphs, SIAM, Jour.of.Alg, Discrete Math. 1 (1980) 382-404. 5| F.Harary, Graph Theory, Addison-Wesley, Reading, Massachussets, USA, 1969. 6] Joseph A.Gallian, A dynamic survey of Graph labeling, The Electronic J. Combinatorics. 16 (2013) 1-298. 7| T.K.Mathew Varkey, Some Graph Theoretic Operations Associated with Graph Labelings, Ph.D Thesis, University of Kerala, 2000. 8] A.Nagarajan, A.Nellai Murugan and A.Subramanian, Near meanness on product graphs, Scientia Magna, Vol.6, 3(2010), 40-49. 9| G.Suresh Singh, Graph Theory - A study of certain Labeling problems, Ph.D Thesis, Uni- versity of Kerala, (1993). International J.Math. Combin. Vol.1(2015), 86-95 The Forcing Vertex Monophonic Number of a Graph P.Titus (University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Nagercoil-629004, India) K.Iyappan (Department of Mathematics, National College of Engineering, Maruthakulam, Tirunelveli-627151, India) E-mail: titusvino@yahoo.com; iyappan_1978@yahoo.co.in Abstract: For any vertex x in a connected graph G of order p > 2, a set Sz C V(G) is an x-monophonic set of G if each vertex v € V(G) lies on an x — y monophonic path for some element y in Sz. The minimum cardinality of an z-monophonic set of G is the z-monophonic number of G and is denoted by mz(G). A subset Tz, of a minimum xz-monophonic set Sz of G is an x-forcing subset for Sz if Sz is the unique minimum x-monophonic set containing Tz. An «-forcing subset for Sz of minimum cardinality is a minimum «-forcing subset of Sz. The forcing x-monophonic number of Sz, denoted by fm,(Sz), is the cardinality of a minimum x-forcing subset for S;. The forcing x-monophonic number of G is fm,(G) = min{ fm, (Sz)}, where the minimum is taken over all minimum x-monophonic sets Sz; in G. We determine bounds for it and find the forcing vertex monophonic number for some special classes of graphs. It is shown that for any three positive integers a, b and c with 2 <a<b<c, there exists a connected graph G such that fm,(G) = a, mz(G) = b and cmz(G) = c for some vertex xz in G, where cmz(G) is the connected z-monophonic number of G. Key Words: monophonic path, vertex monophonic number, forcing vertex monophonic number, connected vertex monophonic number, Smarandachely geodetic k-set, Smaran- dachely hull k-set. AMS(2010): 05C12 §1. Introduction By a graph G = (V, E) we mean a finite undirected connected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. For basic graph theoretic terminology we refer to Harary [6]. For vertices x and y in a connected graph G, the distance d(x, y) is the length of a shortest « — y path in G. An x — y path of length d(, y) is called an x —y geodesic. The neibourhood of a vertex v is the set N(v) consisting of all vertices wu which are adjacent with v. The closed neibourhood of a vertex v is the set N[v] = N(v)U{v}. A vertex v is a simplicial vertex if the subgraph induced by its neighbors is complete. 1Received April 8, 2014, Accepted February 28, 2015. The Forcing Vertex Monophonic Number of a Graph 87 The closed interval I[x,y] consists of all vertices lying on some x — y geodesic of G, while for S CV, I[S] = U I{x,y]. A set S of vertices is a geodetic set if I[S] = V, and the minimum zjyEes cardinality of a geodetic set is the geodetic number g(G). The geodetic number of a graph was introduced in [1,8] and further studied in [2,5]. A geodetic set of cardinality g(G) is called a g—set of G. Generally, for an integer k > 0, a subset S C V is called a Smarandachely geodetic k-set if I[S\JS*] = V and a Smarandachely hull k-set if In(S\JSt) = V for a subset St CV with |St| <k. Let k =0. Then a Smarandachely geodetic 0-set and Smarandachely hull 0-set are nothing else but the geodetic set and hull set, respectively. The concept of vertex geodomination number was introduced in [9] and further studied in [10]. For any vertex x in a connected graph G, a set S of vertices of G is an x-geodominating set of G if each vertex v of G lies on an x — y geodesic in G for some element y in S. The minimum cardinality of an «-geodominating set of G is defined as the x-geodomination number of G and is denoted by g,(G). An 2-geodominating set of cardinality g,(G) is called a g, — set. A chord of a path P is an edge joining any two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of a graph G is a monophonic set of G if each vertex v of G lies on an x — y monophonic path in G for some z,y € S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). The concept of vertex monophonic number was introduced in [11]. For a connected graph G of order p > 2 and a vertex «x of G, a set S; C V(G) is an z-monophonic set of G if each vertex v of G lies on an x — y monophonic path for some element y in S,;. The minimum cardinality of an z-monophonic set of G is defined as the x-monophonic number of G, denoted by m,(G). An a-monophonic set of cardinality m,(G) is called a m, — set of G. The concept of upper vertex monophonic number was introduced in [13]. An z-monophonic set S, is called a minimal x-monophonic set if no proper subset of S; is an x-monophonic set. The upper x-monophonic number, denoted by m*(G), is defined as the minimum cardinality of a minimal z-monophonic set of G. The connected z-monophonic number was introduced and studied in [12]. A connected x-monophonic set of G is an x-monophonic set S, such that the subgraph G([S..] induced by S, is connected. The minimum cardinality of a connected z-monophonic set of G is the connected x-monophonic number of G and is denoted by cm,(G). A connected x-monophonic set of cardinality cm,(G) is called a cm, — set of G. The following theorems will be used in the sequel. Theorem 1.1([11]) Let x be a vertex of a connected graph G. (1) Every simplicial verter of G other than the vertex x (whether x is simplicial vertex or not) belongs to every mz — set; (2) No cut vertex of G belongs to any mz — set. Theorem 1.2((11]) (1) For any vertex x in a cycle Cp(p > 4), mz(Cp) = 1; (2) For the wheel W, = Ky + Cp_-1(p > 5), Ma(Wp) = p—1 or 1 according as x is K, or x is in Cp_1. 88 P.Titus and K.Iyappan Theorem 1.3([11]) For n> 2, mz(Qn) =1 for every vertex x in Qn. Throughout this paper G denotes a connected graph with at least two vertices. §2. Vertex Forcing Subsets in Vertex Monophonic Sets of a Graph Let x be any vertex of a connected graph G. Although G contains a minimum z-monophonic set there are connected graphs which may contain more than one minimum z-monophonic set. For example, the graph G given in Figure 2.1 contains more than one minimum z-monophonic set. For each minimum x-monophonic set S; in a connected graph G there is always some subset T of S; that uniquely determines S; as the minimum x-monophonic set containing T’. Such sets are called ” vertex forcing subsets” and we discuss these sets in this section. Also, forcing concepts have been studied for such diverse parameters in graphs as the geodetic number [3], the domination number [4] and the graph reconstruction number [7]. Definition 2.1 Let x be any verter of a connected graph G and let S, be a minimum «- monophonic set of G. A subset T of S, is called an x-forcing subset for S, if S, is the unique minimum x-monophonic set containing T. An x-forcing subset for S, of minimum cardinality is a minimum «-forcing subset of S,;,. The forcing x-monophonic number of Sz, denoted by fing (Sx), ts the cardinality of a minimum x-forcing subset for S,. The forcing x-monophonic number of G is fm,(G) = min{fm,(Sz)}, where the minimum is taken over all minimum x-monophonic sets Sz in G. Example 2.2 For the graph G given in Figure 2.1, the minimum vertex monophonic sets, the vertex monophonic numbers, the minimum forcing vertex monophonic sets and the forcing vertex monophonic numbers are given in Table 2.1. Vertex Minimum x az-monophonic sets {ry}. {rz} ir, s} {u,r,y}, {u, 7, z}, {u,r, s} Minimum forcing z-monophonic sets whtehis [a (uh teh ts} a a a {u,r, wh, {u,r.y}, (usr, 2} {wy} {2} | or | {uw} {uy}, fu, 2} (wh, fy} {2} 1 Table 2.1 poe : a . |v _| . pw | | = | . = | ee . U v w y z 8 t r The Forcing Vertex Monophonic Number of a Graph 89 y Figure 2.1 Theorem 2.3 For any vertex x in a connected graph G,0< fm,(G) < mz(G). Proof Let x be any vertex of G. It is clear from the definition of fin, (G) that fm, (G) > 0. Let S$, be a minimum z-monophonic set of G. Since fm,(S2) < mz(G) and since fm, (G) = min{ fm, (Sc) : Se is a minimum xz-monophonic set in G}, it follows that fm,(G) < mz(G). Thus 0 < fim, (G) < mz(G). w y u v z t S Figure 2.2 Remark 2.4 The bounds in Theorem 2.3 are sharp. For the graph G given in Figure 2.2, S = {u, z,t} is the unique minimum w-monophonic set of G and the empty set ¢ is the unique minimum w-forcing subset for S. Hence fm,,(G) = 0. Also, for the graph G given in Figure 2.2, S; = {y} and Sj = {z} are the minimum u-monophonic sets of G and so m,,(G) = 1. It is clear that no minimum u-monophonic set is the unique minimum u-monophonic set containing any of its proper subsets. It follows that fm, (G) = 1 and hence fm,(G) = mu(G) = 1. The inequalities in Theorem 2.3 can be strict. For the graph G given in Figure 2.1, m,,(G) = 2 and fim, (G) = 1. Thus 0 < fim, (G) < mu(G). In the following theorem we characterize graphs G for which the bounds in Theorem 2.3 are attained and also graphs for which fm, (G) = 1. Theorem 2.5 Let x be any vertex of a connected graph G. Then 90 P.Titus and K.Iyappan (1) fm,(G) = 0 if and only if G has a unique minimum x-monophonic set; (2) fm, (G) = 1 if and only if G has at least two minimum «-monophonic sets, one of which is @ unique minimum £-monophonic set containing one of its elements, and (3) fm,(G) = mz(G) if and only if no minimum x-monophonic set of G is the unique minimum x-monophonic set containing any of its proper subsets. Definition 2.6 A vertex u in a connected graph G is said to be an x-monophonic vertex if u belongs to every minimum x-monophonic set of G. For the graph G in Figure 2.1, S; = {u,r,y}, Se = {u,r,z} and S3 = {u,r,s} are the minimum v-monophonic sets and so u and r are the v-monophonic vertices of G. In particular, every simplicial vertex of G other than x is an x-monophonic vertex of G. Next theorem follows immediately from the definitions of an x-monophonic vertex and forcing z-monophonic subset of G. Theorem 2.7 Let x be any vertex of a connected graph G and let Fm, be the set of relative complements of the minimum x-forcing subsets in their respective minimum x-monophonic sets in G. Then rer, F is the set of x-monophonic vertices of G. Theorem 2.8 Let x be any vertex of a connected graph G and let M, be the set of all x- monophonic vertices of G. Then0< fim,(G) < mz(G) — |Mz|. Proof Let S, be any minimum xz-monophonic set of G. Then m,(G) = |S;|, Mz C S, and S, is the unique minimum z-monophonic set containing S,,— M, and so fim,(G) < |S: —M-z| = Ma(G) — |M,|. Theorem 2.9 Let x be any vertex of a connected graph G and let S, be any minimum «- monophonic set of G. Then (1) no cut vertex of G belongs to any minimum «-forcing subset of S,; (2) no x-monophonic vertex of G belongs to any minimum «-forcing subset of Sz. Proof (1) Since any minimum z-forcing subset of S, is a subset of S,, the result follows from Theorem 1.1(2). (2) Let v be an z-monophonic vertex of G. Then v belongs to every minimum x-monophonic set of G. Let T C S, be any minimum x-forcing subset for any minimum x-monophonic set 5S, of G. If vu € T, then T’ = T — {v} is a proper subset of T such that S, is the unique minimum x-monophonic set containing T’ so that T’ is an x-forcing subset for S, with |T’| < |T|, which is a contradiction to T a minimum z-forcing subset for S,. Hence v ¢ T. Corollary 2.10 Let x be any vertex of a connected graph G. If G contains k simplicial vertices, then fm,(G) < mz(G) —k+1. Proof This follows from Theorem 1.1(1) and Theorem 2.9(2). Remark 2.11 The bound for fi, (G) in Corollary 2.10 is sharp. For a non-trivial tree T with The Forcing Vertex Monophonic Number of a Graph 91 k, end-vertices, fm,(T) =0 = mz(T) — k+ 1 for any end-vertex x in T. Theorem 2.12 (1) If T is a non-trivial tree, then fm,(T) =0 for every verter x in T; (2) If G is the complete graph, then fm,(G) =0 for every verter x in G. Proof This follows from Theorem 2.9. 0 if p=3,4 Theorem 2.13 For every verter x in the cycle Cp(p > 3), fmz(Cp) = . 1 if p2d Proof Let Cp : u1,U2,°+: ,Up, ur be a cycle of order p > 3. Let x be any vertex in Cp, say © = uy. If p = 3 or 4, then C, has unique minimum z-monophonic set. Then by Theorem 2.5(1), fm,(Cp) = 0. Now, assume that p > 5. Let y be a non-adjacent vertex of x in Cp. Then S, = {y} is a minimum z-monophonic set of C,. Hence C, has more than one minimum x-monophonic set and it follows from Theorem 2.5(1) that fm,(Cp) 4 0. Now it follows from Theorems 1.2(1) and 2.3 that fm,(G) = mz(G) =1. Theorem 2.14 For any verter x in a complete bipartite graph Kmmn(m,n > 2), fm,(Kmn) = 0. Proof Let (Vi, V2) be the bipartition of Km». If « € Vi, then S, = Vi — {x} is the unique minimum z-monophonic set of G and so by Theorem 2.5(1), fm,(G) = 0. If a € Va, then S, = V2 — {x} is the unique minimum z-monophonic set of G and so by Theorem 2.5(1), fim, (G) = 0. Theorem 2.15 (1) If G is the wheel W, = K, + Cp-i(p = 4,5), then fm,(G) = 0 for any vertex x in Wy; (2) If G is the wheel W, = K, + Cp_-1(p = 6), then fm,(G) =0 or 1 according as x is Ky or x is in Cy_1. Proof Let Cp_1 : U1, U2,°+* , Up—1, U1 be a cycle of order p— 1 and let wu be the vertex of Ky. (1) If p = 4 or 5, then G has unique minimum z-monophonic set for any vertex x in G and so by Theorem 2.5(1), fm, (G) = 0. (2) Let p > 6. If = u, then 5S, = {ur,u2,---,Up—i} is the unique minimum z- monophonic set and so by Theorem 2.5(1), fm,(G) = 0. If € V(Cp-1), say © = us, then S; = {u;} (3 < 7% < p—2) is a minimum z-monophonic set of G. Since p > 6, there is more than one minimum x-monophonic set of G. Hence it follows from Theorem 2.5(1) that fm,(G) 4 0. Now it follows from Theorems 1.2(2) and 2.3 that fim, (G) = mz(G) = 1. 0 wf n=2 Theorem 2.16 For any vertex x in the n-cube Qn (n > 2), then fm,(Qn) = : 1 if n>3 Proof If n = 2, then Q, has unique minimum zx-monophonic set for any vertex x in Q,, and so by Theorem 2.5(1), fm, (Qn) = 0. Ifn > 3, then it is easily seen that there is more than one minimum z-monophonic set for any vertex x in Q,. Hence it follows from Theorem 2.5(1) 92 P.Titus and K.Iyappan that fm,(Qn) 4 0. Now it follows from Theorems 1.3 and 2.3 that fm,(Qn) = ™Mz(Qn) = 1. The following theorem gives a realization result for the parameters fim, (G), m2(G) and mz (G). Theorem 2.17 For any three positive integers a, b and c with2 <a<b<c, there exists a connected graph G with fm,(G) =a, mz(G) = b and mi (G) =c for some verter x in G. Proof For each integer i with 1 <i<a-—1, let Fy : uo4, ui, vei, us, be a path of order 4. Let Cg: t,u,v,w,x,y,t be a cycle of order 6. Let H be a graph obtained from F; and Cg by joining the vertex x of Cg to the vertices uo; and u3,; of Fi(1 <i<a-—1). Let G be the graph obtained from H by adding c— a new vertices yj, y2,°-* , Ye—b; V1; V2;°** ;Vb—a and joining each yi(1 <i <c— 6) to both wu and y, and joining each v;(1 < 7 < b— a) with x. The graph G is shown in Figure 2.3. U1,a—1 U2,a—1 U0,a—1 Figure 2.3 Let S = {v1,v2,--- ,Up—a} be the set of all simplicial vertices of G. For 1 < j < a—1, let S; = {u1,;, 2,7}. If 6 = c, then let S, = {u,v,t}. Otherwise, let S, = {u,v}. Now, we observe that a set S, of vertices of G is a m,-set if S;, contains S' and exactly one vertex from each set S;(1 < 7 < a) so that m,(G) > b. Since s. = SU {u, via, vi,2,°+* ;Us1,a—1} is an x-monophonic set of G, we have m;(G) = b. Now, we show that fm,(G) =a. Let S; = SU {u, ui, ui2,-°+ ,U1,a—1} be a mz-set of G and let T;, be a minimum z-forcing subset of $,. Since S' is the set of all z-monophonic vertices of G and by Theorem 2.8, fim, (G) < ma(G) — |S| =a. The Forcing Vertex Monophonic Number of a Graph 93 If |Z,| < a, then there exists a vertex y € S, such that y ¢ Ty. It is clear that y € S; for some j = 1,2,--- ,a, say y = u11. Let S., = (Se — {u1,1}) U {uo1}. Then Si, # S, and S., is also a minimum x-monophonic set of G such that it contains T,, which is a contradiction to T, a minimum 2z-forcing subset of S,. Thus |T,| = a and so fm,(G) = a. Next, we show that m{(G) =c. Let Uz, = SU {uij1, ui,2,°++ 5 Us,a—15t, Y1, Y2s°** + Ye—b}- Clearly U, is a minimal x-monophonic set of G and so m7 (G) > c. Also, it is clear that every minimal x-monophonic set of G contains at most c elements and hence m{(G) < c. Therefore, mi (G)=c. x The following theorem gives a realization for the parameters fim, (G), m2(G) and cm,(G). Theorem 2.18 For any three positive integers a, b and c with2 <a<b<c, there exists a connected graph G with fm,(G) =a, mz(G) = 6b and cm,(G) =c for some verter x in G. Proof We prove this theorem by considering three cases. Case 1. 2<a<b<e. For each integer 7 with 1 <2 < a—1, let Fy : yi, uri, u24,y3 be a path of order 4. Let Po b42 : Yt, Y2,Y3.°°* 5 Ye—-b+2 be a path of order c— b+ 2 and let P : v1, v2, v3 be a path of order 3. Let H, be a graph obtained from F;(1 <i < a—1) and P._»+42 by identifying the vertices y; and y3 of all Fj(1 <i < a—1) and P.-p42. Let Hz be the graph obtained from A, and P by joining the vertex v; of P to the vertex yo of H, and joining the vertex vs of P to the vertex y3 of H;. Let G be the graph obtained from Hz by adding b — a new vertices 21, 22; ++; Zp—-a and joining each z;(1 < i < b—a) with the vertex y-_p42. The graph G is shown in Figure 2.4. Ul,a—1 U2,a—-1 y ¥3 Ya Ys V1 U3 v2 Figure 2.4 Let x = yo and let S = {z, z2,--- , 2p-a} be the set of all simplicial vertices of G. For 1<j<a-1, let S; = {u1,;,u2,;} and let S, = {ve,v3}. Now, we observe that a set S; of 94 P.Titus and K.Iyappan vertices of G is a m,-set if S, contains S and exactly one vertex from each set Si(1 <j<a). Hence m,(G) > b. Since Ss. = SU {ve, wij, vi,2,°** ;U1,a—1} is an z-monophonic set of G with |S.,| = 6, it follows that m;(G) = b. Now, we show that fm,(G) =a. Let S; = SU {v2, ui, Ui,2, ---) U1,a—1} be a mz-set of G and let T;, be a minimum z-forcing subset of $,. Since S' is the set of all z-monophonic vertices of G and by Theorem 2.8, fim, (G) < m2(G) — |S| =a. If |Z,| < a, then there exists a vertex y € S, such that y ¢ Ty. It is clear that y € S; for some j = 1,2,--- ,a, say y = ui. Let S., = (Se — {uia}) U {uo}. Then Si, # Sy and S\, is also a minimum x-monophonic set of G such that it contains T,, which is a contradiction to T, an x-forcing subset of S,,. Thus |T,| = a and so fm, (G) = a. Clearly, SU {v3, ua,1, U2,2,°°* 5 U2,a—15 Y3> Y45°** » Ye—b+2} is the unique minimum connected x-monophonic set of G, we have cm;(G) = c. Case 2. 2<a=b<candc=0+1. Construct the graph Hz in Case 1. Then G = Hp has the desired properties (S is the empty set). Case 3. 2<a=b<candc>0b+2. For each? with 1 <i<a-—1, let Fy: yi, ui, wiz, ys be a path of order 4. Let Po-a41 : Y1,Y2,Y3,°°* + Yc-a+1 be a path of order c— a+ 1 and let Cs : V1, V2, V3, V4, U5,U1 be a cycle of order 5. Let H be a graph obtained from F; and Pe-a41 by identifying the vertices y, and y3 of all Fj(1 <i <a—1) and P.-a41. Let G be the graph obtained from H by identifying the vertex ye_a41 of Pe-a41 and v; of Cs. The graph G is shown in Figure 2.5. Let 7 = ye. U1,a—1 U2,a—1 U5 U4 } V1 Y1 r= 2 Y3 YA Yo-at v2 U3 Figure 2.5 For 1 <j <a-—1, let S; = {u1,;, u2,;} and let S, = {v3, v4}. Now, we observe that a set Sz of vertices of G is a mz-set if S, contains exactly one vertex from each set 5; (1 < j < a) so that mz(G) > a. Since Ss. = {v3,U1,1, U1,2,°'* ,U1,a—1} is an x-monophonic set of G with |.S’.| = a, we have m,(G) =a. The Forcing Vertex Monophonic Number of a Graph 95 Now, we show that fm,(G) =a. Let S, = {v3, ui,1, U1,2,°°* ,U1,a—1} be a mz-set of G and let T, be a minimum z-forcing subset of S,. Then by Theorem 2.3, fm,(G) < mz(G) = a. If |Z,| < a, then there exists a vertex y € S, such that y ¢ Ty. It is clear that y € S; for some j = 1,2,--- ,a, say y = u11. Let S. = (Se — {uia}) U {1.2}. Then Si, # S, and S, is also a minimum x-monophonic set of G such that it contains T,, which is a contradiction to T;, an x-forcing subset of S,. Thus |T,| = a and so fm, (G) = a. Let S = {vo, v3, U2,1, U2,2,°°* »U2,a—1) Y3s Y45°°* »Ye—ati}. It is easily verified that S is a minimum connected x-monophonic set of G and so cmz(G) = c. Problem 2.19 For any three positive integers a, b and c with2<a<b=c, does there exist a connected graph G with fm,(G) = a, mz(G) = b and cm,(G) =c for some vertex x in G? References e F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, 1990. F. Buckley and F. Harary and L.U.Quintas, Extremal results on the geodetic number of a graph, Scientia, A2 (1988), 17-26. 3] G. Chartrand and P. Zhang, The forcing geodetic number of a graph, Discuss. Math. Graph Theory, 19 (1999), 45-58. 4) G. Chartrand, H.Galvas, F. Harary and R.C. Vandell, The forcing domination number of a graph, J. Combin. Math. Combin. Comput., 25(1997), 161-174. 5] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39(1)(2002), 1-6. 6] F. Harary, Graph Theory, Addison-Wesley, 1969. 7| F. Harary and M. Plantholt, The graph reconstruction number, J. Graph Theory, 9 (1985), 451-454. 8] F. Harary, E.Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling, 17(11)(1993), 87-95. 9] A.P. Santhakumaran and P. Titus, Vertex geodomination in graphs, Bulletin of Kerala Mathematics Association, 2(2)(2005), 45-57. 10] A.P. Santhakumaran and P. Titus, On the vertex geodomination number of a graphs, Ars Combinatoria, 1011(2011),137-151. 11] A.P. Santhakumaran and P. Titus, The vertex monophonic number of a graph, Discussiones Mathematicae Graph Theory, 32 (2012) 191-204. 12] P. Titus and K. Iyappan, The connected vertex monophonic number of a graph, Commu- i) nicated. 13] P. Titus and K. Iyappan, The upper vertex monophonic number of a graph, Communicated. International J.Math. Combin. Vol.1(2015), 96-106 Skolem Difference Odd Mean Labeling of H-Graphs P.Sugirtha, R.Vasuki and J. Venkateswari Department of Mathematics, Dr.Sivanthi Aditanar College of Engineering Tiruchendur-628 215, Tamil Nadu, India E-mail: p.sugisamy28@gmail.com, vasukiseharQ@gmail.com, revathi198715@gmail.com Abstract: A graph G with p vertices and q edges is said to have a skolem difference odd mean labeling if there exists an injective function f : V(G) — {1,2,3,--- ,4q — 1} such that the induced map f* : E(G) — {1,3,5,--- , 2g — 1} defined by f*(uwv) = [ ese is a bijection. A graph that admits skolem difference odd mean labeling is called a skolem difference odd mean graph. In this paper, we investigate skolem difference odd mean labeling of some H-graphs. Key Words: Skolem difference odd mean labeling, skolem difference odd mean graph. AMS(2010): 05C78 §1. Introduction Throughout this paper, by a graph we mean a finite, undirected and simple graph. Let G(V, EF) be a graph with p vertices and q edges. For notations and terminology we follow [1]. Path on n vertices is denoted by P,. Ki,m is called a star and is denoted by S,,. The bistar By» is the graph obtained from K by identifying the center vertices of Ky, and Ky, at the end vertices of Kz respectively. The H-graph of a path P,,, denoted by H,, is the graph obtained from two copies of P,, with vertices v1, v2,--+ ,Un and uy, U2,°-:,Un by joining the vertices Undi and Unti if n is odd and the vertices vz4; and uz ifn is even. The corona of a graph G on p vertices v1, V2,:-* ,Up is the graph obtained from G by adding p new vertices U1, U2,-.-,Up and the new edges uv; for 1 < i < p. The corona of G is denoted by G© Kj. The 2-corona of a graph G, denoted by G © $3 is a graph obtained from G by identifying the center vertex of the star Sj at each vertex of G. The disjoint union of two graphs G, and G2 is the graph G U G2 with V(G1 U G2) = V(G1) UV (G2) and E(G) U G2) = E(G)) U E(G2). The concept of mean labeling was introduced and studied by S. Somasundaram and R. Ponraj [5]. Some new families of mean graphs are studied by S.K. Vaidya et al. [6]. Further some more results on mean graphs are discussed in [4,7,8]. A graph G is said to be a mean graph if there exists an injective function f from V(G) to {0,1,2,--- ,q} such that the induced map f* from E(G) to {1,2,3,--- ,q} defined by f*(uv) = [2] is a bijection. 1Received June 4, 2014, Accepted March 2, 2015. Skolem Difference Odd Mean Labeling of H-Graphs 97 In [2], K. Manickam and M. Marudai introduced odd mean labeling of a graph. A graph G is said to be odd mean if there exists an injective function f from V(G) to {0,1,2,3,--- ,2qg—1} ‘ % x — | f(w)t+f such that the induced map f* from E(G) to {1,3,5,--- ,2qg—1} defined by f*(uv) = [A] is a bijection. Some more results on odd mean graphs are discussed in [9,10]. The concept of skolem difference mean labeling was introduced and studied by K. Murugan and A. Subramanian [3]. A graph G = (V, E) with p vertices and q edges is said to have skolem difference mean labeling if it is possible to label the vertices z € V with distinct elements f(z) from 1,2,3,---,p+q in such a way that for each edge e = uv, let f*(e) = efor) and the resulting labels of the edges are distinct and are from 1,2,3,---,qg. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph. The concept of skolem difference odd mean labeling was introduced in [11]. A graph with p vertices and q edges is said to have a skolem difference odd mean labeling if there exists an injective function f : V(G) > {1,2,3,--- ,4q¢— 1} such that the induced map f* : E(G) - {1,3,5,---,2q— 1} defined by f*(uv) = |] is a bijection. A graph that admits a skolem difference odd mean labeling is called a skolem difference odd mean graph. A skolem difference odd mean labeling of B47 is shown in Figure 1. 1 AT Figure 1 In this paper, we prove that the H-graph, corona of a H-graph, 2-corona of a H-graph are skolem difference odd mean graph. Also we prove that union of any two skolem difference odd mean H-graphs is also a skolem difference odd mean graph. §2. Skolem Difference Odd Mean Graphs Theorem 2.1 The H-graph G is a skolem difference odd mean graph. Proof Let v1, v2,°++,Un and uj,Ug,--- ,Un be the vertices of the H-graph G. The graph G has 2n vertices and 2n — 1 edges. 98 P.Sugirtha, R.Vasuki and J. Venkateswari Define f : V(G) — {1,2,3,--- ,4q — 1 = 8n — 5} as follows: 2i — 1, 1<i<nandi is odd f(vi) = as 8n — 21-1, 1<i<nand {iis even 6n — 21-1, if n is odd, 1 <i <n and 17 is odd Plus) 2n+ 21-1, if n is odd, 1 <i <n andi is even Ui) = 2n+ 21-1, if n is even, 1 <i <n andi is odd 6n — 21-1, if n is even, 1 <i <n and 7 is even. For the vertex labeling f, the induced edge labeling f* is given as follows: f* (wivigi) = 4n — 21-1, 1l<i<n-l f* (uuigi) = 2n — 21-1, 1<i<n-1 f (vag ung) =2n—1 if nis odd and oi (va4iug) =2n—1 if nis even. Thus, f is a skolem difference odd mean labeling and hence the H-graph G is a skolem difference odd mean graph. For example, a skolem difference odd mean labeling of H-graphs G; and G2 are shown in Figure 2. 1 27 35 18 1 9 5 23 27 19 31 17 5 13 9 19 23 15 G, = As G2 =: Fg Figure 2 Theorem 2.2 For a H-graph G, G© Ky is a skolem difference odd mean graph. Proof By Theorem 2.1, there exists a skolem difference odd mean labeling f for G. Let Skolem Difference Odd Mean Labeling of H-Graphs 99 U1, V2,°°* 5 Un and U4, U2,°++ ,Un be the vertices of G. Let V(G © Ki) =V(G)U {vj, vg,-..,u,} U {u}, ug, - and E(G © Ky) = E(G) VU {y;vj, uu, 1 <i <n}. saetiat Case 1. n is odd. Define g: V(G© Ky) > {1,2,--- , 16n — 5} as follows: . nti g(v2i-1) = f(vai-1), IS<is< 5 n-1 2 n+1 g(vos) = f(va) + 8n, 1<i< g(uai—1) = f(ua-1) + 8n, 1<i< g(uri) = f (uri) i g(v%¢-1) = 9(un) —4n—-4(i- 1), 1<i< = a 2 / 7 . n—-1 g(vo;) = g(tn-1) t4n +41, 1<i< a 1 g(Ug;,-1) = g(Un) —2n+4(i-1), 1<i< - =a (ug) = g(Un—1) + 2n — A(t — 1), Lets 9 (vivig1) = f (viviga) +4n, L<i<n-1 GO (ustigi1) = f*(ustiga) +4n, 1L<i<n-1 go (ujv,) =4n+1-2i, 1<i<n g (uu) =2n+1-2i1, l<i<n g* (vagatings) =3f* (vngungs) +2. Case 2. 7 is even. Define g: V(G© Ky) > {1,2,3,---+ ,16n — 5} as follows: g(vai—1) = f(vai-1), 1<i< 5 g(vai) = f(vai) +8n, 1l<i< 5 g(ua—1) = f(ua-1), I<i< 5 g(uai) = f(ua)+8n, 1L<i< 5 g(vb;_-1) = g(tn_1) +4n + 6-41, 1<i< wl s 100 P.Sugirtha, R.Vasuki and J. Venkateswari g(v;) = g(un) —4n-24+4i, 1<i< 5 g(U5¢-1) = g(Un—1) + 2(n + 1) — 4(¢ — 1), g(tn) —2(n+1)+4i, 1<i< For the vertex labeling g, the induced edge labeling g* is obtained as follows: Gg (vivigi) = f* (vivian) + 4n G (witiga) = f*(ustsgi) + 4n g* (ujyu;) = 4n +1 - 2% g* (uzuh) = 2n +1—- 2% go” (vgsiug) = 3f* (vgsiug) +2 Thus, g is a skolem difference odd mean labeling and hence G © Ky is a skolem difference odd mean graph. For example, a skolem difference odd mean labeling of H-graphs G1,G2,G, © Ky, and G2 © Ky are shown in Figure 3. 1 13 1 27 43 31 35 13 ! Mo VE 17 39 7 31 17 9 21 2 " 35 23 G, = Hs Go = He 47 49 43 53 39 57 Gio ky Figure 3 Skolem Difference Odd Mean Labeling of H-Graphs 101 Theorem 2.3 For a H-graph G, G© S2 is a skolem difference odd mean graph. Proof By Theorem 2.1, there exists a skolem difference odd mean labeling f for G. Let V1, V2,°°* Un and U1, U2,+++ , Un be the vertices of G. Let V(G) together with vj, v5,--- v1, vf, v%, Ur ul, us: ui, and uy, us,--- , ul form the vertex set of GO $2 and the edge set is E(G) together with {u,vj, vjuy’, uiui, usul) : 1 <i <n}. Case 1. nis odd. Define g: V(G© S2) > {1,2,3,--- ,24n — 5} as follows: g(vai-1) = f(vai1), I<i< St g(va:) = f(va)+16n, 1<i< — u g(uz—1) = f(ua—1) + 16n, 1<i< i = ; g(u2i) = f(ua), 1<i< = : 9(V2i-1) =9(Un) —4n—124i-1), 1l<i< — : g(v5,) = 9(un—-1) +4n-4412i, 1<i< —= . 934-1) = 9(V4-1)-—4, T<is< 2 ~ : (vf) = oleh) +4, 1<a< 2+ g(Ug,-1) = g(un) —6n+12%—-1), 1<i< “ : g(uy;) = g(tn_1) +4n4+ 18-121, 1<i< “; : g(ux-1) = g(ugy_1) +4, 1<i< < : n—-1 g(ux) = 9(Ux%)-4, 1<i< For the vertex labeling g, the induced edge labeling g* is given as follows: g (ViVi41 *(Uivi41) + 8n, 1 < 1 < n—-1 ) 9" (uittizi) = f*(usuigi) +8n, 1<ic<n-1 go (vjv,) =8n+3-4i, 1<i<n nm+1-4i, 1l<i<n 8 8 )=4n+3-44, 1l<i<n g (uu) =4n+1-4i, l<i<n 5 Case 2. nis even. 102 P.Sugirtha, R.Vasuki and J. Venkateswari Define g: V(G© S2) > {1,2,3,--- ,24n — 5} as follows: (vir) = flexi), 1<i<8 g(voi) = f(vai) + 16n, 1<i< 5 g(u2i—1) = f(ua-1), 1<i< ; g(uai) = f(ua:) +16n, L<i< 5 g(uy;) = g(un) —6n-6+4+12i, L<i< 5 7 n g(ug;_1) = g(ugy1)-4, 1<i< 3 g(us;) = g(uh;) +4, 1<i< > i 51 67 21 1 17 5 47 59 43 63 25 5 91 9 43 55 39 oe 29 9 25 13 39 51 35 55 33 13 29 17 35 AT 31 Gy = Hy G2 = Ag Figure 4 Skolem Difference Odd Mean Labeling of H-Graphs 103 For the vertex labeling g, the induced edge labeling g* is obtained as follows: gO (U;vie1) = f* (Vivig1) +8n, L<i<n-1 gO" (ustigia) = f* (uid) +8n, L<i<n-1 g* (vv!) = 8n 43-4, 1<i<n go (uy) = 8n+1-4i, 1<i<n g*(ujul) =4n 43-45, 1<i<n go (uu) =4n+1-4i, Ll<i<n 9° (vesiuz) = 5f* (vgsiug) +4. Thus, f is a skolem difference odd mean labeling and hence the graph G © Sz is a skolem difference odd mean graph. For example, a skolem difference odd mean labeling of H-graphs G,G2,G , © Sz and G2 © S2 are shown in Figures 4 and 5. 143 125 139 2 ares 129 83 79 77 211 94 tat aes - 79 81 137-128 117 131 69 5 191 187 171 141 (4 127 71 73 oe 89 115 207 25 5 91 ie 67 ia 63 ee 149 f 129 8 167 9 187 153 183 133 115 59 85 55 101 203 29 s 103 9 25 7 107 WL gg rm 13 183 ites 179 163 ae 103 47 97 43 113 199 33 ot 13 29 117 43 87 a) 95 17 179 173105 175 159 153 91 177 109 157 G1 © So G2 © So Figure 5 Theorem 2.4 If G, and G2 are skolem difference odd mean H-graphs, then G, U G2 is also a skolem difference odd mean graph. 104 P.Sugirtha, R.Vasuki and J. Venkateswari Proof Let V(G1) = {uwi,u,:1<i <n} and V(G2) = {s,,t; :1< 7 < m} be the vertices of the H-graphs G; and G2 respectively. Then the graph G; U G2 has 2(n + m) vertices and 2(n+m-— 1) edges. Let f : V(G1) > {1,2,3,--- ,8n—5} and g: V(G2) — {1,2,3,--- ,8m—5} be a skolem difference odd mean labeling of G; and G2 respectively. Define h : V(Gy U G2) > {1,2,3,--- ,4qg— 1 = 8(n +m) — 9} as follows: For1<i<nandn>1, Ce f (us) if 7 is odd f(ui) + 8m —-—4 if 7 is even f(vi) + 8m —4 if n is odd and i is odd f (vi) if n is odd and 7 is even h(v;) = f (vi) if n is even and i is odd f(vi) +8m—4 if nis even and i is even. h(s;) = g(s;) +2 h(t;) = g(t;) +2. For the vertex labeling h, the induced edge labeling h* is given as follows: For1<i<n-—landn>1, h* (uptigi) = f*(usuigi) + 4m — 2 h*(v;vi41) = f* (vivi41) + 4m — 2 I ‘i h* (wags vnga 1) = f(u nga Ungs ) + 4m —2 if n is odd and yee h* (ua 4iv2 *(u uny .1v2) +4m—2 if n is even. Forl1<j<m-landm>1, h*(s38j41) = h*(tytj41) = h* (smgat mgs ) = g (smgat mgs ) if m is odd 2 h* (sm4itm) =g (sm4itm) if m is even. Thus, f is a skolem difference odd mean labeling of Gj U Gz and hence the graph GU G2 is a skolem difference odd mean graph. For example, a skolem difference odd mean labeling of G; U Gz where G, = H3; Go = Skolem Difference Odd Mean Labeling of H-Graphs 105 As, Gy = As; Go = Ag, Gy = Ag; Go = Hg and Gy = Ag; Go = Ay are shown in Figures 6-9 following. 1 27 3 29 35 13 37 15 1 15 1 51 5 23 7 25 19 9 55 9 31 17 33 19 5 11 5 7 Cie 9 19 1 21 Go = Hs G, UGy = H3U As Figure 6 1 13 3 15 1 27 = 43 31 1 71 = 45 33 35 13 5 17 79 13 7 19 5 23 39 27 5 67 Al 29 31 17 9 21 75 17 11 23 9 19-35 23 9 63-37 25 G, = Hs; Go = He G, UGp = Hs U Hg Figure 7 1 13 3 15 43 31 45 33 1 9 5 17 1 9 7 19 27 19 39 27 71 63 Al 29 5 13 9 21 5 13 11 23 23 15 35 23 67 59 37 25 G, = 4 Go = He G, UGg = H,U Hg Figure 8 106 P.Sugirtha, R.Vasuki and J. Venkateswari 1 9 | 9 1 9 8 11 27 19: 27 19 55 47-99 21 5 3 5 13 5 1367 15 23 15 23 15 51 43 25 awe G1 = 4 G2 = Hy, G, UGp = H4U 4 Figure 9 References F.Harary, Graph Theory, Addison-Wesley, Reading Mass., 1972. K.Manikam and M.Marudai, Odd mean labeling of graphs, Bulletin of Pure and Applied Sciences, 25E(1) (2006), 149-153. K.Murugan and A.Subramanian, Skolem difference mean labeling of H-graphs, Interna- tional Journal of Mathematics and Soft Computing, 1(1) (2011), 115-129. A.Nagarajan and R.Vasuki, On the meanness of arbitrary path super subdivision of paths, Australas. J. Combin., 51(2011), 41-48. S.Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science Letters, 26(2003), 210-213. S.K.Vaidya and Lekha Bijukumar, Some new familes of mean graphs, Journal of Mathe- matics Research, 2(3) (2010), 169-176. R.Vasuki and A.Nagarajan, Meanness of the graphs P,,, and P?, International Journal of Applied Mathematics, 22(4) (2009), 663-675. R.Vasuki and S.Arokiaraj, On mean graphs, International Journal of Mathematical Com- binatorics, 3(2013), 22-34. R.Vasuki and A.Nagarajan, Odd mean labeling of the graphs P,,,, P? and Peas Kragujevac Journal of Mathematics, 36(1) (2012), 125-134. R.Vasuki and S.Arokiaraj, On odd mean graphs, Journal of Discrete Mathematical Sciences and Cryptography, (To appear). R.Vasuki, J. Venkateswari and G.Pooranam, Skolem difference odd mean labeling of graphs (Communicated). International J.Math. Combin. Vol.1(2015), 107-112 Equitable Total Coloring of Some Graphs Girija.G (Department of Mathematics, Government Arts College, Coimbatore - 641 018, Tamil Nadu, India) Veninstine Vivik.J (Department of Mathematics, Karunya University, Coimbatore 641 114, Tamil Nadu, India) E-mail: prof_giri@yahoo.co.in; vivikjose@gmail.com Abstract: In this paper we determine the equitable total chromatic number yer for the double star graph Kin, and the fan graph Finn. Key Words: Equitable total coloring, double star graph, fan graph. AMS(2010): 05C15, 05069 §1. Introduction In this paper, we consider only finite simple graphs without loops or multiple edges. Let G(V, FE) be a graph with the set of vertices V and the edge set E. Total coloring y;(G) was introduced by Vizing [7] and Behzad [1]. They both conjectured that for any graph G the following inequality holds: A(G) + 1 < x4(G) < A(G) +2. It is obvious that A(G) + 1 is the best possible lower bound. This conjecture is proved so far for some specific classes of graphs. In general the equitable total coloring problem is more difficult than the total coloring problem. In 1994, Fu [4] gave the concepts of an equitable total coloring and the equitable total chromatic number of a graph. For a simple graph G(V, E), let f be a proper k—total coloring of G [Zs] — (ZG |) <1, 4.9 = 1,2,...5k. The partition {T;} = {V; UE; :1<7< k} is called a k—equitable total coloring (k-—ETC of G in brief), and Xet (G) = min {k|k — ETC of G} is called the equitable total chromatic number [2-6, 10] of G, where Vz € T; = Vi UE, f (x) = i, i= 1,2,---,k. It is obvious that ye(G) > A+ 1. Furthermore Fu presented a conjecture concerning the equitable total chromatic number (simply denoted by ETCC) Conjecture 1.1([4]) For any simple graph G(V, E), Xet(G) < A(G) + 2. 1Received September 3, 2014, Accepted March 8, 2015. 108 Girija.G and Veninstine Vivik.J These Researchers in [2, 3, 5, 6, 8-10] have concentrated in providing the equitable total chromatic number for specific families of graphs. Lemma 1.2([10]) For any simple graph G(V, E), Xet (G) 2 xe(G) 2 A(G) +1. Lemma 1.3([4]) For complete graph K, with order p, Dp, p = 1mod 2 Xet (Kp) a pt+1, p=Omod 2. Lemma 1.4([3]) For n > 13 the total equitable chromatic number of Hypo-Mycielski Graph, Xer(HM(W,,)) =n + 2. In [9], equitable total chromatic numbers of some join graphs were given. Gong Kun et.al [2] proved some results on the equitable total chromatic number of Wy, V Kn, Fm V Ky and Sm V Ky. In 2012, Ma Gang and Ma Ming [6] proved some results concerning the equtiable total chromatic number of Py, V Sy, Pm V Fy and Pin V Wr. In the present paper, we find the equitable total chromatic number yz for the double star graph Ky,n,, and the fan graph Fyn. §2. Preliminaries Definition 2.1 A double star Ky, is a tree obtained from the star Ky, by adding a new pen- dant edge of the existing n pendant vertices. It has 2n+1 vertices and 2n edges. Let V (Kinin) = {vu} U {u1, v2,.-., Un} U {u1, U2,---,Un} and E (Kinin) = {e1, €2,---, en} U {1, $2,--+, Sn}. Definition 2.2 A Fan graph Ky, + Py where Py is path on n vertices. All the vertices of the fan corresponding to the path P,, are labeled from m to n consecutively. The vertices in the fan corresponding Km is labeled m+n. §3. Equitable Total Coloring of Double Star Graphs Theorem 3.1 For any positive integer n, Xet (Kinin) =n+l. Proof Let V(Kinn) = {uo} U {u:l <i <nbU {yu :1l<i<n} and E(Kinn) = {e,: 1 <i<n}U{s;:1<i< n}, where e; (1 <i < n) is the edge upu; (1 <i <n) and s,(1 <i <n) is the edge u,v; (1 < i < n). Now we partition the edge and vertex sets in K1,n,» as follows. Equitable Total Coloring of Some Graphs 109 T, = {€1, Un, $n—1, Un—2} Tz = {€2,U1,8n;Un—1} Tz; = {€3, U2, Un} Ts = {e4,u3, 51} Tr = {€k, Up—1, $k—3, Ura} (5 Sk <n) Tnti = {U0,$n—2;Un—3} Clearly T,,72,7T3,T4,T, and T,41 are independent sets of Kin». Also |Zi| = |T2| = |T,| = 4(5 <k <n) and |T3| = |T1| = |Tn41| = 3, it holds the inequality ||T;| — |T;|| < 1 for every pair (i,j). This implies yer (Kinn) < n+1. Since the set of edges {€1, €2,...€n} and uo receives distinct color, Yet (Kinjn) > Xt (Kann) > n+1. Hence Yer (Ki.nn) > n+1. Therefore Xet (Kann) =n+ ale §4. Equitable Total Coloring of Fan Graphs Theorem 4.1 For any positive integer n,m (n > m), then A+2ifn=m A+2ifn-m=2 A+lifn-m=1 n+2ifn—-m>3 Xet (Bm,n) = Proof Let V (Fmjn) = {ui:1<i<m}U{u;:1<j<nband FE (Finn) = U {eng :1 <7 < n}u i=1 {ej:1<j<n-—1}, where e; (1 <j < n—1) is the edge vjvj41 (1 <j<n =) and e;,; is the edge ujvj (1<i<m,1<j<n). Now we partition the edge and vertex sets of Fin, in the following cases. Case l. n=m Ti T» T3 Ta—4 Ta—3 I l| l| I 1611} OY ean, Cond, CORA) oss ema} Uist 161 5,91} UY es ns Ce Verna) emo Pua} {€1,3, €2,2; €3,1} U {€6,n; €7,n—15 €8,n—2)+++,€m,6} U {us} {e1,n—2; €2,n-3)+++5 €m—2,1} U {Un} {e1n-1, C2 n—-2Qye 285 €m—1,1} 110 Girija.G and Veninstine Vivik.J Txse 2S * NEG Cinaissiny Cpt} Ta-1 = {e2,n; €3,n—-1) ae eng €m,2} U {vi} Ta = {e3,n; €4,n—-1) ee | €m,3} U {v2} . n—-1 ; n Tayi = jeaxr:l<ix< oe U {usin 1<i< [=]; : n—-1 : n Tayo = 9je€a-1:1<i< ae Ufuii:isis |e|t Clearly T,, T2, T3,...Ta+2 are independent sets of Fyn. It holds the inequality ||T;| — |T;|| < 1 for every pair (7,7). This implies yer (Fim) < A+ 2. Each edge vjvj41 (1 <j <n—1) is adjacent with 2m + 2 edges and incident with two vertices,it forms a triangle with atleast one vertex of {uj:1<%<m}. Therefore equitable total coloring needs A + 2 colors.xet (Finn) > xt (Fm) => A+ 2. Hence yer (Finn) > A+ 2. Therefore xet (Fm) = A+ 2. Case 2. n-—m=2 T, = feri}U fe2,n; €3,n—1; €4,n—2;-+++,€m,a} U {v3} Tz = {€1,2,€2,1} U {€3,n; €4n—15 €5,n—2;+-+)€m,5} U {va} T3 = {€1,3, €2,2,€3,1} U {€4,n; €5,n—1, €6,n—2)++-,€m,6} U {v5} Ta-2 = {€1,n—2,€2,n—3,-+-3€m,i$ U {Un} Ta-1 = {€1,n-1, €2,n—2,---;€m2}U {ri} TA = {€1jn,€2,n-1;--++€m,3} U {vo} —1 —1 Tav1 = fuasisis |" *]bufesisis|% \} —2 —1 Tava = fuistsis |" Juferasis |] \} Clearly T,, T2, T3,...Ta+2 are independent sets of Fiy,,n. It holds the inequality ||T;| — |T;|| < 1 for every pair (7,7). This implies yer (Fim) < A+2. Since each edge v;v;41 (1 <j <n-1) is adjacent with 2m-+ 2 edges and incident with two vertices,it forms a triangle with atleast one vertex of {u;:1<i< m}. Therefore equitable total coloring needs A + 2 colors. xet (Fmjn) > Xt (Finn) > A+ 2. Hence Ver (Finn) > A+ 2. Therefore yer (Finjn) = A+ 2. Case 3. n-—m=1 Ti = {e1,1}U {€2.n, €3,n—-1, €4n—2,-++; m3} U {v2} Tz = {€1,2,€2,1} U {€3,n, €4n—1; €5,n—2;---, €m,4} U {v3} Ts) = {e1,5,.c9;9, €3,1} U {edn; C8, nois C6,nid) 0.9m} Uva} Ta-2 = Ta-1 l| Ta I I Ta+i Clearly Ti, Ta, Ts, ays Equitable Total Coloring of Some Graphs 111 (Oi iy Co otee.< perma | LR} {eta Conese {vi} ots [52 oft [2 owns BSH fofotses 2H . Ta+1 are independent sets of Finn. It holds the inequality ||T;| — |T;|| < 1 for every pair (i,j). This implies yet (Finn) < A+ 1. Since at each vertex v; (2 <j <n-1) there exist A mutually adjacent edges and v; (2 < 7 < n— 1) needs one more color. Yer (Fimjn) > Xt (Finn) > A+1. Hence ver (Finn) > A+1. Therefore yer (Finn) = At 1. Case 4. n-—m>3 Tl = To = T3 = Tn-2 ae. Th-1 = T, = Th41 7 Tn+42 — Clearly Ti, Ta, Ts, ol {eri} U {€2,n, €3,n—1 €4,n—2;+++,€m,4} U {vs} {e1,2, €2,1} U {€3,n, €4,n—1, €5,n—2)-++3€m,5} U {us} {e1,3, €2,2, 3,1} U {€4,n; €5,n—1; €6,n—2,-++3€m,6} U {us } {€1,n—2; €2,n—3)--++;€m a} U {un} {€1,n-1, €2,n—2,-+-,€m,2}U {ui} {€1,n; €2,n—-1;+++;€m,3} U {v2} fuiaitsis[Z]}ufen:isis eS] fux:isis|Z)}uf{enaisis [4] Tn+2 are independent sets of Fyn. It holds the inequality ||T;| — |T;|| < 1 for every pair (i,j). This implies yet (Finn) < 2+ 2. Since at each vertex u; (1 <i < m) there exist n mutually adjacent edges and u;(1 <i <m) needs one more color. Vet (Finn) => Xt (Finn) > m+ 2. Hence xXet (Fin) > n+ 2. Therefore xet (Fm) = n+ 2. References [1 [2] Behzad.M, Graphs and Their Chromatic Numbers, Doctoral Thesis, East Lansing: Michi- gan State University, 1965. Gong Kun, Zhang Zhongfu, Wang Jian Fang, Equitable total coloring of some join graphs, Journal of Mathematical Research & Exposition, 28(4), (2008), 823-828. DOI:10.3770/j.issn: 1000-341X.2008.04.010 Haiying Wang, Jianxin Wei, The equitable total chromatic number of the graph HM(W,,), J. Appl. Math. & Computing, Vol. 24(1-2), (2007), 313-323. Hung-lin Fu, Some results on equalized total coloring, Congr. Numer., 102, (1994), 111- 112 [10 Girija.G and Veninstine Vivik.J 119. Ma Gang, Ma Ming, The equitable total chromatic number of some join-graphs, Open Journal of Applied Sciences, (2012), 96-99. Ma Gang, Zhang Zhongfu, On the equitable total coloring of multiple join-graph, Journal of Mathematical Research and Exposition, 2007. 27(2), 351-354. Vizing.V.G, On an estimate of the chromatic class of a p-graph(in Russian), M etody Diskret. Analiz., 3, (1964), 25-30. Wei-fan Wang, Equitable total coloring of graphs with maximum degree 3, Graphs Combin., 18, (2002) 677-685. Zhang Zhongfu, Wang Weifan, Bau Sheng. et al. On the equitable total colorings of some join graphs, J. Info. & Comput. Sci., 2(4), (2005), 829-834. Zhang Zhongfu, Zhang Jianxun, Wang Jianfang, The total chromatic number of some graph, Sci. Sinica, Ser.A, 31(12), (1988) 1434-1441. International J.Math. Combin. Vol.1(2015), 113-125 Some Characterizations for the Involute Curves in Dual Space Stleyman SENYURT ( Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52100, Ordu/Turkey) Mustafa BILICI (Department of Mathematics, Faculty of Education, Ondokuz Mayis University, Turkey) Mustafa CALISKAN (Department of Mathematics, Faculty of Science, Gazi University, Turkey) E-mail: senyurtsuleyman@hotmail.com, mbilici@omu.edu.tr, mustafacaliskan@gazi.edu.tr Abstract: In this paper, we investigate some characterizations of involute — evolute curves in dual space. Then the relationships between dual Frenet frame and Darboux vectors of these curves are found. Key Words: Dual curve, involute, evolute, dual space. AMS(2010): 53A04, 45F10 §1. Introduction Involute-evolute curve couple was originally defined by Christian Huygens in 1668. In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve. In particular, the involute of a given curve is a well known concept in the classical differential geometry (for the details see [7]). For classical and basic treatments of Involute-evolute curve couple, we refer to [1], [5], [7-9] and [13]. The relationships between the Frenet frames of the involute-evolute curve couple have been found as depend on the angle between binormal vector B and Darboux vector W of evolute curve, [1]. In the light of the existing literature, similar studies have been constructed on Lorentz and Dual Lorentz space,[2-4, 10-12]. In this paper, The relationships between dual Frenet frame and Darboux vectors of these curves have been found. Additionally, some important results concerning these curves are given. §2. Preliminaries Dual numbers were introduced by W.K. Clifford (1849-79) as a tool for his geometrical investi- 1Received July 2, 2014, Accepted March 10, 2015. 114 Siileyman SENYURT, Mustafa BILICI and Mustafa GALISKAN gations. The set [D = {A=a+ea* | a,a*eIR, e* =0} is called dual numbers set. On this set product and addition operations are described as (a+ea*)+(b+eb*) = (a+b)+e(a*+0*), (a+ ca*).(b+b*) ab + « (ab* +. a*b), l Saad respectively. The elements of the set [D? = 4 A| A= a +ea*, a,a*eIR®} are called dual vectors. On this set, addition and scalar product operations are described as ® : ID¢x ID? + ID? , A@B= (a+ i) re(ai +i), Or = POST? ID? 3 oA= rd be (20! +xa), respectively. Algebraic construction (I D3,6,ID,+, ©) is a modul. This modul is called ID-Modul. The inner product and vectorel product of dual vectors A, BeID® are defined by respec- Ch) oT Tp SID: (4,8) = (4,0) +2((a.s*) + (#3) A: ID?xID3 ID, AnB=(aa BJ re(Tas rata b). tively, For A¥<0, the norm [A of A= @ +a" is defined by a la)= (2.4) =| rear [alee | The angle between unit dual vectors A and B ® = y+ ey” is called dual angle and this angle is denoted by ((6]) — A, B) = cos® = cosy — ey*siny Let ao &# “Pe LR Tp? s — a(s) =a(s)+¢ea*(s) be differential unit speed dual curve in dual space [D?. Denote by {T, N, B} the moving dual Frenet frame along the dual space curve a (s) in the dual space [D?. Then T, N and B are the dual tangent, the dual principal normal and the dual binormal vector fields, respectively. The function « (s) = k, + ekt and 7 (s) = kg + kd are called dual curvature and dual torsion of a, Some Characterizations for the Involute Curves in Dual Space 115 respectively. Then for the dual curve a the Frenet formulae are given by, , T (s) = #«(s)N(s) N'(s) = —K(s)T(s)+7(s) B(s) (2.1) B (s) = -1(s)N(s) The formulae (2.1) are called the Frenet formulae of dual curve. In this palace curvature and torsion are calculated by, det (7. qe T’) ee a Re 2 2.2 n= YP.) 1()= eam (2.2) If a is not unit speed curve, then curvature and torsion are calculated by lo’ (s)Aa’ (s)| det (a (s),a" (s),a” (s)) s) = ; 3 , T(s)= 7 7 5 (2.3) Ilo" (s)|| Ila’ (8) Aa” (s)|| By separating formulas (2.1) into real and dual part, we obtain a (s) = kin n(s) = —kit+keb (2.4) b (s) = —kon ee (s) = kyn*+kin n* (s) = —kyt* — kit + kob* + kb (2.5) bY (s) = —kyn* — kon §3. Some Characterizations Involute of Dual Curves Definition 3.1 Let @: I = ID? and 8: I > ID® be dual unit speed curves. If the tangent lines of the dual curve a is orthogonal to the tangent lines of the dual curve 3 , the dual curve B is called involute of the dual curve a or the dual curve a is called evolute of the dual curve 3 (see Fig.1). According to this definition, if the tangent of the dual curve a is denoted by T and the tangent of the dual curve B is denoted by T, we can write a) =0 (3.1) Theorem 3.1 Let a and B be dual curves. If the dual curve B involute of the dual curve a, we can write B(s) =a(s)+[(a — 8) tec] T (s) , C1, coeIR. 116 Siileyman SENYURT, Mustafa BILICI and Mustafa GALISKAN Proof Then by the definition we can assume that B(s) =&(s) +AT(s)_, d(s) = p(s) en" (s) (3.2) for some function A(s). By taking derivative of the equation (3.2) with respect to s and applying the Frenet formulae (2.1) we have dp dX where s and s* are arc parameter of the dual curves @ and B , respectively. It follows that pe at SON een (3.3) ds ds the inner product of (3.3) with T is a" (rr) =(149)\ersany (3.4) ds ds From the definition of the involute-evolute curve couple, we can write (1,7) =0 By substituting the last equation in (3.4) we get 14+ a “ana < GOES (3.5) Straightforward computation gives i" (s) =—1 and Vi (s) =0 integrating last equation, we get p(s) =c.—s and p*(s)=c (3.6) By substituting (3.6) in (3.2), we get B(s) — &(s) = [(c1 — 8) + €c2] T (s). (3.7) This completes the proof. Corollary 3.1 The distance between the dual curves 8 and @ is \c, — s| Fec2. Proof By taking the norm of the equation (3.7) we get d (a (s) ,3(s)) Lithy alse (3.8) Some Characterizations for the Involute Curves in Dual Space 117 This completes the proof. Hi Fig.1 Theorem 3.2 Let a, B be dual curves. If the dual curve B involute of the dual curve a, then the relationships between the dual Frenet vectors of the dual curves @ and 3 = N = —cos®7'+ sin®B wi Sy = sin@T+cos®B Proof By differentiating the equation (3.2) with respect to s we obtain B' (s) = AK (s)N(s) , A= (cr — 8) tECo (3.9) and T = N(s) (3.10) 118 Siileyman SENYURT, Mustafa BILICI and Mustafa GALISKAN By differentiating the equation (3.9) with respect to s we obtain ~ GB =—dAK*T4 (rx n) N+ «7B If the cross product fed A B. is calculated we have BAB =XK2rT +2 B (3.11) The norm of vector p A cm is found RAB avevier er (3.12) Bi A B’ For the dual binormal vector of the dual curve B we can write Bl A B" pes By substituting (3.11) and (3.12) in the last equation we get B= os + as (3.13) For the dual principal normal vector of the dual curve B we can write N=BAT and Ne 2a ia (3.14) ST + —— B VK2+ 72 VK24+ 72 Let ® (® =ptey* ,ct7= 0) be dual angle between the dual Darboux vector W of @ and dual unit binormal vector B in this situation we can write K sin® = cos® = Pome (3.15) Kept? By substituting (3.15) in (3.12) and (3.13) the proof is completed. The real and dual parts of ie N, B are ip et. N = —cos@T +sin®@B B = sin®T+cosOB is separated into the real and dual part, we can obtain Some Characterizations for the Involute Curves in Dual Space 119 t= n, n = —cosyt + singh, be sinyt + cospb * + | lI n n = —cosyt* + sinyb* + y* (singt + cosyb) —— sinyt* + cosypb* + y* (cosyt — sinyb) On the way sin® = sin (py + ey*) = siny + ey* cosy cos® = cos (yp + ey*) = cosp — ey*siny If the equation is separated into the real and dual part, we can obtain ke inp = k2 + kS — Qhy kok — 2k2ks y (ke + kB)” cosp = If the equation is separated into the real and dual part, we can obtain cos = ue e RR Gene Qk? + kX + Qkikgks — k2k* — k3ks yo (k2 + 3)? Theorem 3.3 Let a, 8 be dual curves. If the dual curve B involute of the dual curve a , curvature and torsion of the dual curve 2 are 2, n2(s)-+72(s) * ae '' "ORO Ole) oy) Proof By the definition of involute we can write B(s) =&(s) + |A|T(s) (3.17) 120 Siileyman SENYURT, Mustafa BILICI and Mustafa GALISKAN By differentiating the equation (3.17) with respect to s we obtain df ds* Ze ds = T+ TO+A KONO, dp ds* Se = P)-T)+IAR(8) NOs), (5) = = [Alm (s) N(s). Since the direction of T (s) is coincident with N (s) we have Taking the inner product of (3.18) with T and necessary operation are made we get ds* = [X(s)| (5). By taking derivative of (3.19) and applying the Frenet formulae ( 2.1) we have , = —_ d. 2k T (s) =N(s)=T (s) 7 = —«T +7B. From (3.20) and (3.21), we have =" —KT+7TB From the last equation we can write = = = —KT + TB BIN OE) Taking the inner product the last equation with each other we have (HOMORONO) = (Raper ROHR) Thus, we find =a Ge K? (s) +7? (s)_ A? (8) K? (s) We know that BING = d2K2rT + d2K3B. Taking the norm the last equation, we get Bin a" | Shen (ee ery (3.18) (3.19) (3.20) (3.21) Some Characterizations for the Involute Curves in Dual Space 121 By substituting these equations in (2.3), we get 0 KA 0 —K?X (KA) KTX = (=n?) —kK (KA) —K3yX+ (KA) — KT7X (KA) T+ (rd) — ———; ‘ | B' A B" = KT —KT T= k |A| (K? + 77) This completes the proof. If the equation (3.16) is separated into the real and dual part, we can obtain i = Viki + k5 pk,’ 2k JRE +E : e kik, — kok, kg a pay? ky (k? + k8) _* (kik + kyk* — ky kk — kak; ) ko => a (pk? + ky k3 11) [2 (kik® + kok) kup + (k2 + h3) (ku + heap) (Fak, = kak) (wh? + ky k3y)” Theorem 3.4 Let a, 3 be dual curves and the dual curve B involute of the dual curve a . If W and W are Darbous vectors of the dual curves @ and B we can write 1 W => (w +0) (3.22) Proof Since W is Darboux vector of 3 (s) we can write W (s) =7 (s)T(s) + «(s) B(s) (3.23) By substituting 7, a kK, B in the last equation, we get t Ci = KT —KT Ke +7? W (s) = PARE eee + nl (sin®T + cos®B) . (3.24) 122 Siileyman SENYURT, Mustafa BILICI and Mustafa CALISKAN By substituting (3.15) in (3.24), we get W (s) KT —KT n() +2" (TS) S) So. Ss — === 7 KIN GB +7) KA Vt The necessary operation are made, we get = TT +B KT —KT —— ———————“ V/ W@= a tapes ey a 1 KT KT W (s) aM (7+ xo+ Sen) and ne (2)! x? Wea NO aaa Furthermore, Since sn® 7t/V/K2+7?7 cosh fn? +72" Lee tan®. K By taking derivative of the last equation, we have ® sec? = (=) : K By a straightforward calculation, we get , T ’ K = (gts K/ K2 +47? ae (w+e'n) 7 RD which completes the proof. If the equation (3.22) is separated into the real and dual part, we can obtain = wtyn vw = ——, bk _* pk, (w* +9'n4 p*n) (uk; 4 wk) (w+¢'n) W ——d ————— yerky If the equation (3.24) is separated into the real and dual part, we can obtain = k? + k3 w= ve (sinyt + cospb) , [ky Some Characterizations for the Involute Curves in Dual Space 123 a V/ ke + he ‘ * * * : OS ae (sinyt™ + cospb* + y* (cosyt — sinyb)) LRy juky (kiki + kok3) — (ki +3) (uki + wks) Ty ak? (sinyt + cosy) - Theorem 3.5 Let &, B be dual curves and the dual curve B involute of the dual curvea . If C and C are unit vectors of the direction of W and W, respectively Do V2 + 72 CS = (3.25) Proof Since B the dual angle between W and B we can write C (s) =sin@T (s) + cos@B (s) . In here, we want to find the statements sin and cos, we know that ~ T T sing = = = x | + 41 By substituting tT and « in the last equation and necessary operations are made, we get , ® VO24 624 72 sing = (3.26) Similarly, Poe af D2 + KA = Q = [j-2 2 Oe ee VJ O'2 + 62 + 72 VO’? + 624 72 which completes the proof. cos = (3.27) Thus we find If the equation (3.25) is separated into the real and dual part, we can obtain - ent Ske + kge or p +kit+ kg ; - 2 2 kik +koks ge n( REIS) o( vo" +brkj +a) * pnt te’ nt (Bae + beitee, OOO yp +k? + ke 124 Siileyman SENYURT, Mustafa BILICI and Mustafa CGALISKAN If the equation (3.26) and (3.27) are separated into the real and dual part, we can obtain , sinp = SS = : Vp tkhi+ks _ (® +P +P )O*—o'p + hikt + kokdy 8) SS SSS Ss 3 2 a a eae £ ki + kg cosp = >, . yp” +k? + k2 ee A ? (ee tH haky + hoks) VIPER — (o" +82 +48) (ky + kok) sing = @ (®2 462472)? (M+R Corollary 3.2 Let a, 8 be dual curves and the dual curve B involute of the dual curve a . If evolute curve @ is helix, (1) The vectors W and B of the involute curve B are linearly dependent; (2) C=C; (3) @ is planar. Proof (1) If the evolute curve @ is helix, then we have a , — =tan® = cons or ® =0 K and then we have sind = 0, cosb = 1. Thus, we get ®=0. (3.28) (2) Substituting by the equation (3.28) into the equation (3.25), we have C=C. (3) For being is a helix , then we have - = cons, (=) = 0. (3.29) Some Characterizations for the Involute Curves in Dual Space 125 On the other hand, from the equation (3.16), we can write which completes the proof. Gas Brcaeadl = AF) al (3.30) K (R27? )2 (Guanes ta esa Substituting by the equation (3.29) into the equation (3.30), then we find 7 =0, References 1 (11 [12 [13] Bilici M. and Caliskan M., Some characterizations for the pair of involute-evolute curves in Euclidean space, Bulletin of Pure and Applied Sciences, Vol.21E, No.2, 289-294, 2002. Bilici M. and Caliskan M., On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space, International Mathematical Forum, Vol. 4, No.31, 1497-1509, 2009. Bilici M. and Caliskan M., Some new notes on the involutes of the timelike curves in Minkowski3-space, Int.J.Contemp.Math. Sciences, Vol.6, No.41, 2019-2030, 2011. Bikcii B and Karacan M.K., On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3 space, Int. J. Contemp. Math. Sciences, Vol. 2, No. 5, 221 - 232, 2007. Fenchel W., On the differential geometry of closed space curves, Bull. Amer. Math. Soc., Vol.57, No.1, 44-54, 1951. Hacisalihoglu H. H., Acceleration Axes in Spatial Kinematics I, Communications, Série A: Mathématiques, Physique et Astronomie, Tome 20 A, pp. 1-15, Année 1971. Hacisalihoglu H.H., Differential Geometry, (Turkish) Ankara University of Faculty of Sci- ence, 2000. Millman R.S. and Parker G.D., Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977. Sabuncuoglu, A., Differential Geometry (Turkish), Nobel Publishing, 2006. Senyurt S. and Giir S., On the dual spacelike-spacelike involute-evolute curve couple on dual Lorentzian space, International Journal of Mathematical Engineering and Science, 1s8n:2277-6982, vol.1, Issue :5, 14-29, 2012. Senyurt S. and Giir S., Timelike - spacelike involute - evolute curve couple on dual Lorentzian space, J. Math. Comput. Sci., Vol.2, No. 6, 1808-1823, 2012. Senyurt S. and Gtir S., Spacelike - timelike involute- evolute curve couple on dual Lorentzian space, J. Math. Comput. Sci., Vol.3, No.4,1054-1075, 2013. Yiice S. and Bektas O., Special involute-evolute partner D-curves in E° , European Journal of Pure and Applied Mathematics, Vol. 6, No. 1, 20-29, 2013. International J.Math. Combin. Vol.1(2015), 126-135 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs V.Ramachandran (Department of Mathematics, P.S.R Engineering College, Sivakasi, Tamil Nadu, India) C.Sekar (Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, Tamil Nadu, India) E-mail: me.ram111@gmail.com; sekar.acas@gmail.com Abstract: A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function ¢ from the vertex set of G to {0,1, N,(N+1),2N, (2N+1),---,N(q—- 1), N(q—1)+1}in such a way that (i) ¢ is 1—1 (ci) ¢@ induces a bijection ¢* from the edge set of G to {1,N+1,2N+1,--- ,N(q—1)+1}where ¢*(uv)=|¢(u) — ¢(v)|. In this paper we prove that arbitrary supersubdivision of disconnected path and cycle P, UC; is one modulo N graceful for all positive integer N. Also we prove that the graph P7 — vy) is one modulo N graceful for every positive integer N. Key Words: Graceful, modulo N graceful, disconnected graphs, arbitrary supersubdivi- sion graphs, Pn J Cn and Pt — vw, AMS(2010): 05C78 §1. Introduction S. W. Golomb [3] introduced graceful labelling. Odd gracefulness was introduced by R. B. Gnanajothi [4]. C. Sekar [11] introduced one modulo three graceful labelling. In [8,9], we introduced the concept of one modulo N graceful where N is any positive integer. In the case N = 2, the labelling is odd graceful and in the case N = 1 the labelling is graceful. Joseph A. Gallian [2] surveyed numerous graph labelling methods. Recently G. Sethuraman and P. Selvaraju [5] have introduced a new method of construction called supersubdivision of a graph. Let G be a graph with n vertices and t edges. A graph H is said to be a supersubdivision of G if H is obtained by replacing every edge e; of G by the complete bipartite graph K2, for some positive integer m in such a way that the ends of e; are merged with the two vertices part of Kom after removing the edge e; from G. A supersubdivision H of a graph G is said to be an arbitrary supersubdivision of the graph G if every edge of G is replaced by an arbitrary Kom (m may vary for each edge arbitrarily). A graph G is said to be connected if any two vertices of G are joined by a path. Otherwise it is called disconnected graph. G. Sethuraman and P. Selvaraju [6] proved that every connected graph has some supersub- 1Received July 15, 2014, Accepted March 12, 2015. One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 127 division that is graceful. They pose the question as to whether some supersubdivision is valid for disconnected graphs. [10] We proved that an arbitrary supersubdivision of disconnected paths are graceful. Barrientos and Barrientos [1] proved that any disconnected graph has a supersubdivision that admits an a-labeling. They also proved that every supersubdivision of a connected graph admits an a-labeling. In this paper we prove that arbitrary supersubdivision of disconnected path and cycle P,, UC; is one modulo N graceful for all positive integer VN. When N = 1 we get an affirmative answer for their question. Also we prove that the graph P+ — oi) is one modulo N graceful for every positive integer N. §2. Main Results Definition 2.1 A graph G with q edges is said to be one modulo N graceful (where N is a positive integer) if there is a function ¢ from the vertex set of G to {0,1,N,(N+1),2N,(2N + 1),...,N(q@—1), N(q—1) +1} in such a way that (2) ¢ is 1 — 1 (it) @ induces a bijection ¢* from the edge set of G to {1,N+1,2N+1,...,N(q—1)+1}where ¢* (uv) =|¢(u) — o(v)]. Definition 2.2 In the complete bipartite graph Kom we call the part consisting of two vertices, the 2-vertices part of Kom and the part consisting of m vertices the m-vertices part of Kom.Let G be a graph with p vertices and q edges. A graph H is said to be a supersubdivision of G if HT is obtained by replacing every edge e of G by the complete bipartite graph Kom for some positive integer m in such a way that the ends of e are merged with the two vertices part of Kom after removing the edge e from G. H is denoted by SS(G). Definition 2.3 A supersubdivision H of a graph G is said to be an arbitrary supersubdivision of the graph G if every edge of G is replaced by an arbitrary Kom (m may vary for each edge arbitrarily). H is denoted by AS'S(G). Definition 2.4 Let v1, v2,...,Un be the vertices of a path of length n and vw), ws), rae, yb) be the pendant vertices attached with v1, v2,...,Un respectively. The removal of a pendant vertex yd) where 1< k<n from Pr yields the graph Pt — ou), Theorem 2.5 Arbitrary supersubdivision of disconnected path and cycle P, UC; is one modulo N graceful provided the arbitrary supersubdivision is obtained by replacing each edge of G by Kam with m > 2. Proof Let P, be a path with successive vertices v1, U2,-+- ,Un and let e; (1 <i <n-—1) denote the edge v;v;41 of P,. Let C, be a cycle with successive vertices Un41, Un+25°°* 5 Un-+r and let e;(n +1<i<n+r) denote the edge vjv;41. Let H be an arbitrary supersubdivision of the disconnected graph P,,UC;, where each edge e; of P, UC; is replaced by a complete bipartite graph Ko, with m; > 2 for l<i<n-1 andn+1<i<n+r. Here the edge Un4;Un+41 is replaced by k2,--1. We observe that H has M = 2(my + me +--+ + Mp1 + Magi $:++ + Mn+r) edges. 128 V.Ramachandran and C.Sekar 1 of} (1) Vi2 a AN (2) (mi) “2 (3) (ma) 8 U1 i = Vig =Vi12 V93 = V93 Figure 1 Supersubdivision of P; U C3 Define o(vu;) = NG —1), += 1,2,3,--- ,n, o(v;) = N(i), i=n+1,n+2,n4+3,---,n+r, and for k = 1,2,3,...,mi, N(M—2k+1)+1 ifi=1 N(M —2+4)+1—2N(m, + mg+---+my1+k—1) iff =2,3,...n—-1 o(u.,)=% N(M—14%) +1-2N(m t+met---+ mp1 tk-1) ifi=ntl N(M —1+%)+1-—2N[(m, + m2 +--+ +mn-1)+ (Mng1 te+e $mG-1) +k -1) ifi=n+2,n4+3,...n4+r—-1 and for k = 1,2,3,--+,mntr, (0) 41) =N(nt+r—k+mnyr) +1 From the definition of ¢ it is clear that {o(vi),é=1,2,--- nt+r} L{o?,,),6=1,2,---,n+7—-Land k = 1,2,3,--- mab {oe nti) b= 1,2,3,-++ mi} = {0,N,2N,---,N(n-1)} {Nin +1), N(n + 2),---,N(n+r)} J{N[M - 2k + 1) +1, N[M — 2m] +1, N[M - 2m - 2] +1---, N[M — 2(my + m2) + 2] +1, N[M — 2(m + me) +1) +1, N[M — 2(m, + me2)—- 1) 4+1,...,N[M — 2(m, + m2 4+ m3) + 3) +1, ..., N[M —34+n—-2(m,+m2+--++mn-2)| +1, N[M —5+n—2(my+me+---+mp_-2)] +1,..., N[M —1+n—2(my+m2+-+++mn-1)] +1, N[M +n — 2(m, + mg +++ mMn—1)| +1, N[M +n —2(m + mg +--+ +mMn—-1+1)) +1,..., N[M +n—2(my + me +--+ + mp1 + mMn4i — 1] +1, N[M +1+n—- 2(m + me +--+ + mMn-1 + Mn41)] +1, One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 129 N[M —1+n—2(my + me +--+ mp1 + ™Mn4i)] +1,-°-, N[M+34+n-— 2(my + me +--+ + mp1 + Mn4 Mn+2)) +1, N[M +2+n—2(mi + me +--+ + mMn-1+ Magi + Mn42)] +1, N[M + n= 2(my + mg +--+ + Mp1 + Mn41 + Mn42)] +1,.-., N[M +44+n—2(mi + me +-+++mn-1 t+ mnt + Mnt2 + Mn+3)] +1, N[M —-24+n+r—2(m, + me +--+ + mp1 + Mn41 + Mn42)] +1, N[M —24+n+r —2[(my + me2+++++ mp1) + (Mngi + Mnpe +++ Mntr—2)]] +1, N{[M —44+n+7—2[(my + me +--+ + mn—1) + (Mngi + M42 ++++ + Mn+r—2)]) +1, -,N[M+n+r—2[(m + mg +-+++mn-1) + (Magi + Mnpe +++ + Mntr—1)]] + 1} LJ{iN(ntr-14+ mare) +1,N(nt+r—2+mngr) +1,--- ,N(n+r)+1} Thus it is clear that the vertices have distinct labels. Therefore ¢ is 1 — 1. We compute the edge labels as follows: For k = 1,2,-++ ma, o*(v{'3v1) =| o(vt}) — o(rr) | = N(M = 2k +1) +1, o*(vfQv2) | $(v{'}) — $(v2) | = N(M — 2k) +1. For k = 1,2,-++,m; and i = 2,3,---,n—1, o*(o,,v;) =| o(v,,) — d(v) | = N(M 2k+1)—2N(m1+m2+---+mi_-1) +1, o* (ve), 0:41) =| (ut) — b(viti) | = N(M — 2k) — 2N(m, +rTmMg+rs': + mji-1) +1. For k= 1,2, mMngas O* (Uta meant) =| Ops neo) — O(Un41) | =N(M — 2k +1) — 2N (my +ima +--+ + mn-1) +1, Omer nga%mt2) =| OUnp 142) ~ O(On42) | = N(M — 2k) — 2N(m, mg cee Mn-1) +1. For k = 1,2, oe MY and j =n+ 2,n =F 3, ore sa+ Tr, o* (uh vi) =| o(u.,) 7 (vi) | = N(M—2k+1)—2N{(mitmgt::-+mMn—-1)+(MagitmMnagat:+-+mi-1)} +1, 6° (ue). vi41) =| o(u), .)—b(viga) | = N(M—2k)-2.N{ (rate +--+ mp1) + (nga tinged bmi-a)} +1. For k= 1,2,-++ mages O (Uo agitate) =| $0) net) — Omer) | = Nasr —k) +1, * k k 6 (UD engrtnsi) =| OO) ns) — OOnt1) | = N(ingr tk - V4. It is clear from the above labelling that the m;+2 vertices of Kom, have distinct labels and the 2m, edges of Kom, also have distinct labels for 1 <<71<n—landn+1<i<n+r-1. Therefore the vertices of each Kom,,1<i<n—-—landn+1<7t<n+r-—1 in the arbitrary supersubdivision H of P,UC; have distinct labels and also the edges of each Kom,,1<%i<n-1 andn+1<i<n+r-—1 in the arbitrary supersubdivision graph H of P, UC, have distinct labels. Clearly H is one modulo N graceful. Hence arbitrary supersubdivisions of disconnected path and cycle P,, UC; is one modulo N graceful, for every positive integer N. Consequently, every disconnected graph has some supersubdivision that is one modulo N graceful. Example 2.6 A odd graceful labelling of AS'S(P3 U C4) is shown in Figure 2. 130 V.Ramachandran and C.Sekar 12 27 14 eC 53 19 59 A aN aS a 8 O55 2 45 i AT Figure 2 Example 2.7 A graceful labelling of AS'S(P3U C3) is shown in Figure 3. 6 8 13 23 26 Li] ° 4 0 24 1 19 2 20 Figure 3 Theorem 2.8 For any pendant vertex oh) € V(P), the graph Pt — od) is one modulo N graceful for every positive integer N. One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 131 Proof Let v1,v2,--- ,Un be the vertices of a path of length n and vt ) ws), tee ese the pendant vertices attached with v1, v2,--- , Un respectively. Consider the graph P* — ol), where 1< k<n. It has 2n —1 vertices and 2n — 2 edges. Case 1. nis even and k is even Define N(2n—3)+1-—2N(i—1) fori=1,2,---, N(2n — 3) +1-2N(£ -1)-N-2N(i- ( (v2i-1) = b(v2i) = N(2i—1) fori = 1,2,--- , 8, (oy = 2N(n—2)+1-—2N(i-—1) fori=1,2,---,£-1 2N(n— 2) +1—2N(% — 2)-3N—-2N(i- (£ +1)) fori= $+1,842,---, NI5 o(v?.,) = 2N(é—1) for i =1,2,-- From the definition of ¢ it is clear that are {d(vai-1), t= 1,2, ee Mace t= 1, 2, oy 5} Oy ie k_4k if Bien uke Ut?) t= poled list 2eth5} cannes 4=1,2,--- 5} = {N(2n—3) +1, N(Qn—5)4+1,---,N(Qn-—k-1)4+1,N(Qn-—k-2)+1 N(Q2n—k—4)+1,...,Nn+1} U{N,3N,---,N(n—1)} | {2N (nm — 2) +1,2N(n— 3) +1,---,N(Qn—k)+1,N(2n—k-3) +1 N(2n—k—5)+1,-+-,N(n—1) +1} LJ{0,2N,..., N(n— 2)} Thus it is clear that the vertices have distinct labels. Therefore ¢ is 1 — 1. We compute the edge labels as follows. For i = 1,2,--+ , 8, $*(v2i—1v2i)=| O(vai—1) — (vai) |= N(2n — 4i) +1, 6* (voi—10§)) 1) = | (v1) — oo 5 |= N(2n — 44 +1) +1. For i = 1, 2, peekaer! E_1, om (Voi4102:) =| (v2i41) — b(v2:) |= N(2n—4i—2)+1, om (vo v24) = | o(v?) = b(va¢) |= N(2n — 4¢- 1) +1. For i = & + 1,4 + 2,° so sae * (Voi-1V2i) =| (vai 1) — H(v24) = Bue — 41+ 1) +1 * (v0i—10$;) 1 )=| $(v2i—1) — O(OR21) [= N(2n — 46+ 2) +1, 6*(v§? v21)=] (04?) — $(021) I= N(2n — 4i) +1. Fori=&41,$42,---, 2-1, b*(v2i¢1v21)=| O(veigi) — O(vai) | = N(Qn - 41-1) +1. This show that the edges have the distinct labels {1,N +1,2N+1,---, N(q—1)+ 1}, where g = 2n — 2. Hence for every positive integer N, P+ — oy) is one modulo N graceful if n is even and k is even. 132 V.Ramachandran and C.Sekar Example 2.9 A one modulo 10 graceful labelling of Pi — vy is shown in Figure 4. 171 10 151 30 131 50 121 70 101 90 0 161 20 141 40 60 111 80 91 Figure 4 Case 2. n is even and k is odd Define . . k=1 N(2%—1) fori=1,2,--- ,= = N(k—2)+N+2N(i—(S2)) for i= St, 8... ,8 (vai) = @(vai-1) = N(2n 3) +1 2N(i 1) fori =1,2,--- 5 (1) 2N(i—1) fori=1,2,---,4+ O(v3;-1) = ) 2N(455+ -1)+3N +2N(i— (48)) fori= 52,52)... 8 gol) = 2N(n — 2) +1-—2N(é-1) for i= 1,2,---, 3. The proof is similar to that of Case 1. Hence for every positive integer N, P* — yd) is one modulo N graceful if n is even and k is odd. Example 2.10 A one modulo 4 graceful labelling of Pi — vy) is shown in Figure 5. 85 4 77 #12 69 «=6©200661- 853 32 45 40 0 81 8 73 16°. 65-24. of 49 36 41 Figure 5 Case 3. nis odd and k is even Define One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 133 z bo 3 w —o = 1) = Nil e ey 1) Zz bad . sabe. a ~_ NI pole ne aa = Ss = = ll NI 4 = i=} fi os (v2) = N(2i — 1) foraH12) sige | an 2N(n—2)+1-—2N(i—1) fori=1,2,...,8-1 2N(n — 2) +1—2N(# — 2) —3N —2N(i— ( (usp 4) =2N(i—1) for i= 1,2,..., 254. From the definition of ¢ it is clear that , n—1 ; n—-1 {(vai-1), tw 12.0 ’ 9 }U{¢(o2), t= 1, 25 5 2 } ? k k k n-1 : n—-1 Liters), t=1,2,--- ie ig aig ea SI} ULo 2), ¢=1,2,---, i = {N(2n— 3) +1,N(2n—5)+1,...,N(Qn—k-1)+1,N(2n—k—-2) +1, N(2n—k-4)+1,...,N(n—1) +1} UJ{N,3N,...,N(n— 2)} {2n(n - 2) +.1,2N(n— 3) +1,...,N(Qn— hk) +1, N(2n-k- 3) +1, N(2n—k-5)+1,...,Nn+1} |}{0,2N,...,N(n—1)} Thus it is clear that the vertices have distinct labels. Therefore ¢ is 1 — 1. We compute the edge labels as follows: Fort =1,2,---, E, Q* (vai—-1¥2i1) =| O(vai-1) — O(vas) | = N(Qn — 42) + 1, o* (voi_1v9) 4) = | $(vai—1) — o(wS1) | = N(2n = 4641) +1. For? =1,2,---, E_1, O* (voi41V2i1)=| 6(vait1) — b(vai) | = N(2n—4i—2)4+1, o* (vs) vai) — | oS?) — b(vai) | = N(Qn-4i-1) $1. For i= $+1,$4+2,---, 354, b*(vai-1v2)=| O(v2i-1) — O(vx) | = N(Qn - 46+ 1) +1, $* (of) ure) = | (vg?) — O(a) | = N(2n- 44) +1. For i= $+1,$42,---, 254, b* (vaigivai) = | O(v2ig1) — 6(v2«) | = N(Qn- 4¢- 1) +1. For i= $41, 8-42,---, 982, d*(va—r0gy1) = | $(vai—1)— d(vgy)1) | = N(Qn— 4442) +1. This show that the edges have the distinct labels {1,N +1,2N+4+1,---, N(q—1) +1}, where g = 2n — 2. Hence for every positive integer N, P+ — yd) is one modulo N graceful if n is odd and k is even. Example 2.11 A one modulo 3 graceful labelling of Pi — vs) is shown in Figure 6. 134 V.Ramachandran and C.Sekar 70 3 67 39 61 15 55 21 49 27) 43-33 37 0 6 64 12 58 18 52 24 46 30 40 36 Figure 6 Case 4. nis odd and k is odd Define N(2i—1) fori=1,2,---,4+ b(vai) = + RED) N(k-2)+N+2N(G-(S5 )) fori = KHL. kt3.... 2 b(vai-1) = N(2n — 3) +1-—2N(i-1) fori =1,2,...,3, 2N(i—1) fori=1,2,---,%54 1 1“) ’ (v3.4) = 2 ; 2N (4+ —-1)+3N 4+ 2N(i— (48)) fori = 42, 58,... 8H oop) =2N(n — 2) +1—2N(i—1) for i =1,2,..., 252. The proof is similar to that of Case 3. Hence for every positive integer N, P* — yl) is one modulo N graceful if n is odd and k is odd. Example 2.12 A one modulo 5 graceful labelling of Pi} — vs) is shown in Figure 7. 96 5 86 15 76 20 66 30 56 40 46 Figure 7 §3. One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 135 Conclusion Subdivision or supersubdivision or arbitrary supersubdivision of certain graphs which are not graceful may be graceful. The method adopted in making a graph one modulo N graceful will provide a new approach to have graceful labelling of graphs and it will be helpful to attack standard conjectures and unsolved open problems. References 1 C. Barrientos and S. Barrientos, On graceful supersubdivisions of graphs, Bull. Inst. Combin. Appl., 70 (2014) 77-85. Joseph A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Com- binatorics, 18 (2011), #DS6. S.W.Golomb, How to number a graph in Graph theory and computing R.C. Read, ed., Academic press, New York (1972)23-27. R. B. Gnanajothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, 1991. G. Sethuraman and P. Selvaraju, Gracefulness of arbitrary supersubdivisions of graphs, Indian J. Pure Appl. Math., 32 (2001) 1059-1064. G. Sethuraman and P. Selvaraju, Super-subdivisions of connected graphs are graceful, preprint. Z. Liang, On the gracefulness of the graph C,, U P,, Ars Combin., 62(2002), 273-280. V. Ramachandran, C. Sekar, One modulo N gracefullness of arbitrary supersubdivisions of graphs, International J. Math. Combin., Vol.2 (2014) 36-46. V. Ramachandran, C. Sekar, One modulo N gracefulness of supersubdivision of ladder, Journal of Discrete Mathematical Sciences and Cryptography (Accepted). C. Sekar and V. Ramachandren, Graceful labelling of arbitrary supersubdivision of discon- nected graph, Ultra Scientist, 25(2)A (2013) 315-318. C. Sekar, Studies in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, 2002. International J.Math. Combin. Vol.1(2015), 186-136 AMCA ~— An International Academy Has Been Established W.Barbara (Academy of Mathematical Combinatorics & Applications, Colorado, USA) The Academy of Mathematical Combinatorics & Appli- cations (AMCA), initiated by mathematicians of USA, China and India in December of last year has been es- tablished in USA. Its aim is promoting the progress and upholding the development of combinatorics and Smarandache multispaces with mathematical sciences, i.e., mathematical combinatorics, including algebra, topology, geometry, differential equations, theoretical physics, theoretical chemistry and mathematical theory on recycling economy or environmental sciences, for instance, the industrial ecology, and so as to advance a sustainable developing for global economy. The activities of AMCA include organizing meetings of the membership, discussions on important topics, edit and publish scientific journals, special volumes and members monographs, awards and honours, recognizing mathematical talent, improvement of mathematics education and taking up other issues concerning to mathematical science. 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Skolem Difference Odd Mean Labeling of H-Graphs By P.SUGIRTHA, R.VASUKI, J.VENKATESWARI Equitable Total Coloring of Some Graphs By GIRIJA G, V.VIVIK J Some Characterizations for the Involute Curves in Dual Space By SULEYMAN SENYURT, MUSTAFA BILICI, MUSTAFA CALISKAN One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs By V.RAMACHANDRAN, C.SEKAR AMCA - An International Academy Has Been Established W.BARBARA An International Journal on Mathematical Combinatorics