Some Star and Bistar Related Divisor Cordial Graphs

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1 Annals o Pure and Appled Mathematcs Vol. 3 No ISSN: X (P) (onlne) Publshed on 3 May 03 Annals o Some Star and Bstar Related Dvsor Cordal Graphs S. K. Vadya and N. H. Shah Department o Mathematcs Saurashtra Unversty Rajkot Gujarat Inda E-mal: samrkvadya@yahoo.co.n Government Polytechnc Rajkot Gujarat Inda E-mal: nrav.hs@gmal.com Receved 9 May 03; accepted 7 May 03 Abstract. A dvsor cordal labelng o a graph G wth vertex set V s a bjecton rom V to {... V } such that an edge uv s assgned the label ether ( u) ( v) or () v () u and the label 0 ( u) ( v) then number o edges labeled wth 0 and the number o edges labeled wth der by at most. A graph wth a dvsor cordal labelng s called a dvsor cordal graph. In ths paper we prove that splttng graphs o star Kn and bstar B nn are dvsor cordal graphs. Moreover we show that degree splttng graph o B nn shadow graph o Bnn and square graph o B nn admt dvsor cordal labelng.. Keywords: Dvsor cordal labelng star bstar. AMS Mathematcs Subject Classcaton (00): 05C78. Introducton We begn wth smple nte connected and undrected graph G = ( V( G) E( G)) wth p vertces and q edges. For standard termnology and notatons related to graph theory we reer to Gross and Yellen [] whle or number theory we reer to Burton []. We wll provde bre summary o dentons and other normaton whch are necessary or the present nvestgatons. Denton.. I the vertces are assgned values subject to certan condton(s) then t s known as graph labelng. Any graph labelng wll have the ollowng three common characterstcs:. A set o numbers rom whch vertex labels are chosen;. A rule that assgns a value to each edge; 67

2 S. K. Vadya and N. H. Shah 3. A condton that ths value has to satsy. Accordng to Beneke and Hegde [3] graph labelng serves as a ronter between number theory and structure o graphs. Graph labelngs have many applcatons wthn mathematcs as well as to several areas o computer scence and communcaton networks. Accordng to Graham and Sloane [4] the harmonous labelngs are closely related to problems n error correctng codes whle odd harmonous labelng s useul to solve undetermned equatons as descrbed by Lang and Ba [5]. The optmal lnear arrangement concern to wrng network problems n electrcal engneerng and placement problems n producton engneerng can be ormalsed as a graph labelng problem as stated by Yegnanaryanan and Vadhyanathan [6]. For a dynamc survey o varous graph labelng problems along wth an extensve bblography we reer to Gallan [7]. Denton.. A mappng : V( G) {0} s called bnary vertex labelng o G and () v s called the label o the vertex v o G under. Notaton.3. I or an edge e= uv the nduced edge labelng * : E( G) {0} s gven by * () e = () u () v. Then v ( ) = number o vertces o G havng label under e ( ) = number o vertces o G havng label under * Denton.4. A bnary vertex labelng o a graph G s called a cordal labelng v (0) v() and e (0) e (). A graph G s cordal t admts cordal labelng. The concept o cordal labelng was ntroduced by Caht [8]. Ths concept s explored by many researchers lke Andar et al. [90] Vadya and Dan [] and Nasreen []. Motvated through the concept o cordal labelng Babujee and Shobana [3] ntroduced the concepts o cordal languages and cordal numbers. Some labelng schemes are also ntroduced wth mnor varatons n cordal theme. Product cordal labelng total product cordal labelng and prme cordal labelng are among menton a ew. The present work s ocused on dvsor cordal labelng. Denton.5. A prme cordal labelng o a graph G wth vertex set VG ( ) s a bjecton : V( G) {3 V( G) } and the nduced uncton * : E( G) {0} s dened by *( e = uv) = gcd( ( u) ( v)) = ; = 0 otherwse. whch satses the condton e (0) e (). A graph whch admts prme cordal labelng s called a prme cordal graph. The concept o prme cordal labelng was ntroduced by Sundaram et al. [4] and n the same paper they have nvestgated several results on prme cordal labelng. Vadya and Vhol [56] as well as Vadya and Shah [78] have proved many results on prme cordal labelng. 68

3 Some Star and Bstar Related Dvsor Cordal Graphs Motvated by the concept o prme cordal labelng Varatharajan et al. [9] ntroduced a new concept called dvsor cordal labelng by combnng the dvsblty concept n Number theory and Cordal labelng concept n Graph labelng. Ths s dened as ollows. Denton.6. Let G = ( V( G) E( G)) be a smple graph and : {... V( G) } be a bjecton. For each edge uv assgn the label ether ( u) ( v) or () v () u and the label 0 ( u) ( v). s called a dvsor cordal labelng e (0) e (). A graph wth a dvsor cordal labelng s called a dvsor cordal graph. In the same paper [9] they have proved that path cycle wheel star Kn and K3n are dvsor cordal graphs whle Kn s not dvsor cordal or n 7. Same authors n [0] have dscussed dvsor cordal labelng o ull bnary tree as well as some star related graphs. It s mportant to note that prme cordal labelng and dvsor cordal labelng are two ndependent concepts. A graph may possess one or both o these propertes or nether as exhbted below. ) P n (n 6) s both prme cordal as proved n [4] and dvsor cordal as proved n [9]. ) C 3 s not prme cordal as proved n [4] but t s dvsor cordal as proved n [9]. ) We ound that a 7-regular graph wth vertces admts prme cordal labelng but does not admt dvsor cordal labelng. v) Complete graph K 7 s not a prme cordal as stated n Gallan [7] and not dvsor cordal as proved n [9]. Generally there are three types o problems that can be consdered n the area o graph labelng.. How a partcular labelng s aected under varous graph operatons;. To nvestgate new graph amles whch admt partcular graph labelng; 3. Gven a graph theoretc property P characterze the class/classes o graphs wth property P that admt partcular graph labelng. The problems o second type are largely dscussed whle the problems o rst and thrd types are not so oten but they are o great mportance. The present work s amed to dscuss the problems o rst knd n the context o dvsor cordal labelng. Denton.7. For a graph G the splttng graph S ( G) o a graph G s obtaned by addng a new vertex v ' correspondng to each vertex v o G such that Nv () = Nv ('). Denton.8. [] Let G = ( V( G) E( G)) be a graph wth V = S S S S T 3 where each S s a set o vertces havng at least two vertces o the same degree and T = V S. The degree splttng graph o G denoted by DS( G ) s obtaned rom G by addng vertces w w w3 wt and jonng to each vertex o S or t. 69

4 S. K. Vadya and N. H. Shah Denton.9. The shadow graph D ( G) o a connected graph G s constructed by takng two copes o G say G and G ''. Jon each vertex u n G to the neghbours o the correspondng vertex v n G. Denton.0. For a smple connected graph G the square o graph G s denoted by G and dened as the graph wth the same vertex set as o G and two vertces are adjacent n G they are at a dstance or apart ng.. Man Results Theorem.. S ( K n) s a dvsor cordal graph. Proo : Let v v v 3 vn be the pendant vertces and v be the apex vertex o Kn and uu u u3 un are added vertces correspondng to vv v v3 vn to obtan S ( K n). Let G be the graph S ( K n) then VG ( ) = n+ and E( G) = 3n. To dene : V( G) { n+ } we consder ollowng three cases. Case : n= to 8 For n= (v)=4 (u)= (v )=3 (v )= and (u )=5 (u )=6. Then e (0) = 3 = e (). For n=3 (v)=3 (u)= (v )=5 (v )=6 (v 3 )=7 and (u )= (u )=9 (u 3 )=4. Then e (0) = 5 e () = 4. For n=4 (v)=3 (u)= (v )=5 (v )=6 (v 3 )=8 (v 4 )=0 and (u )= (u )=4 (u 3 )=7 (u 4 )=9. Then e (0) = 6 = e (). For n=5 (v)=3 (u)= (v )=5 (v )=6 (v 3 )=8 (v 4 )=0 (v 5 )= and (u )= (u )=4 (u 3 )=7 (u 4 )=9 (u 5 )=. Then e (0) = 7 e () = 8. For n=6 (v)=3 (u)= (v )=5 (v )=6 (v 3 )=8 (v 4 )=0 (v 5 )= (v 6 )=4 and (u )= (u )=4 (u 3 )=7 (u 4 )=9 (u 5 )= (u 6 )=3. Then e (0) = 9 = e (). For n=7 (v)=3 (u)= (v )=5 (v )=6 (v 3 )=8 (v 4 )=0 (v 5 )= (v 6 )=4 (v 7 )=6 and (u )= (u )=4 (u 3 )=7 (u 4 )=9 (u 5 )= (u 6 )=3 (u 7 )=5. Then e (0) = 0 e () =. For n=8 (v)=3 (u)= (v )= (v )=6 (v 3 )= (v 4 )=5 (v 5 )=7 (v 6 )= (v 7 )=3 (v 8 )=7 and (u )=4 (u )=8 (u 3 )=0 (u 4 )=4 (u 5 )=6 (u 6 )=8 (u 7 )=9 (u 7 )=5. Then e (0) = = e (). Now or the remanng two cases let n+ n+ 3n s = m = s t = ( n+ s+ ). 3 3 x = s+ t+ x = n x x = n s+ m t x = n x Case : n=9 0 (t=0) 70

5 Some Star and Bstar Related Dvsor Cordal Graphs () v = () u = 3 ( v ) = ( v ) = 6 ; s + x ( vx ) ++ = 5+ 6 ; 0 < x ( vx ) + + = 7+ 6 ; 0 < n s ( u + ) = 4+ 6 ; 0 < n s ( u+ ) = 8+ 6 ; 0 < u = 9+ 6( ); m ( ) n s+ For the vertces ( ux 3+ ) ( ux 3+ ) ( un) assgn dstnct remanng odd numbers. Ths assgns all the vertex labels or case. Case 3: n 3 ( t ) () v = () u = 3 ( v ) = ( v ) = 6 ; s + ( ) v = 9+ 6( ); t s++ x ( vx ) ++ = 5+ 6 ; 0 < x ( vx ) + + = 7+ 6 ; 0 < n s ( u + ) = 4+ 6 ; 0 < n s ( u+ ) = 8+ 6 ; 0 < u = ( v ) + 6 ; m t ( ) n s+ s+ t+ For the vertces ( u ) ( u ) ( u ) assgn dstnct remanng odd numbers. Ths x3+ x3+ n assgns all the vertex labels or case 3. In vew o the above labelng pattern () v ( u ) () v ( u ) n and ( v ) ( v) ( v ) ( u ). Moreover s v v v v v v 3 +s () ( ) () ( ) () ( ) ( u) ( v ) ( u) ( v ) ( u) ( v ) 3 + s ( u) ( v ) ( u) ( v ) ( u) ( v ). and s+ s+ 3 + s+ t 7

6 S. K. Vadya and N. H. Shah 3n Hence e () = n s+ + s+ s+ t = n+ s+ + ( n+ S + ). 3n Thereore n last two case e () = and 3n e (0) =. Thus n all the cases we have e (0) e (). Hence S ( K n) s a dvsor cordal graph. Example.. Let G = S ( K ) VG ( ) = 8 and EG ( ) = 39. In accordance wth 3 Theorem. we have s = 4 m = 4 t = x = 6 x = 7 x = x = and usng the 3 4 labelng pattern descrbed n case. The correspondng dvsor cordal labelng s shown n Fgure. It s easy to vsualze that e (0) = 9 and e () = 0. Fgure : Dvsor cordal labelng o G = S ( K ) 3 Theorem.3. S ( B nn ) s a dvsor cordal graph. Proo: Consder B nn wth vertex set { uvu v n} where u v are pendant vertces. In order to obtan S ( B nn ) add u v u v vertces correspondng to uvu v where n. I G = S ( B nn ) then VG ( ) = 4( n+ ) and EG ( ) = 6n+ 3. We dene vertex labelng : V( G) { 4( n+ )} as ollows. Let p be the hghest prme number < 4( n + ). ( u) = ( u ) = () v = 4 ( v ) = p ( u ) = 6+ ( ); n u = ( u ) + ; n ( ) n For the vertces v v vn and v v v n we assgn dstnct odd numbers (except p ). 7

7 Some Star and Bstar Related Dvsor Cordal Graphs In vew o the above labelng pattern we have e (0) = 3n+ e () = 3n+. Thus e (0) e (). Hence S ( B nn ) s a dvsor cordal graph. Illustraton.4. Dvsor cordal labelng o the graph S ( B66) s shown n Fgure. Fgure : Dvsor cordal labelng o the graph S ( B66) Theorem.5. DS( Bnn ) s a dvsor cordal graph. Proo: Consder B nn wth V( Bnn ) = { u v u v: n} where u v are pendant vertces. Here V( Bnn ) = V V where V = { u v : n} andv = {} u v. Now n order to obtan DS( Bnn ) rom G we add w w correspondng to V V. Then V( DS( B ) = n+ 4and nn E( DS( B )) = { uv uw vw } { uu vv w u w v : n} so EDSB ( ( nn nn ) = 4n+ 3. We dene vertex labelng : V( DS( Bnn )) { n+ 4} as ollows. ( u ) = 4 () v = n+ 3 ( w ) = ( w ) = ( u ) = 3+ ( ); n ( v ) = 6+ ( ); n In vew o the above dened labelng patten we have e (0) = n+ e () = n+. Thus e (0) e (). Hence DS( Bnn ) s a dvsor cordal graph. Illustraton.6. Dvsor cordal labelng o the graph DS( B55) s shown n Fgure 3. 73

8 S. K. Vadya and N. H. Shah Fgure 3: Dvsor cordal labelng o the graph DS( B 55) Theorem.7. D ( Bnn ) s a dvsor cordal graph. Proo: Consder two copes o B nn. Let { uvu v n} and { u v u v n} be the correspondng vertex sets o each copy o B nn. Let G be the graph D ( Bnn ) then VG ( ) = 4( n+ ) and EG ( ) = 4(n+ ). We dene vertex labelng : V(G) { 4(n+)} as ollows. Let p be the hghest prme number and p be the second hghest prme number such that p < p < 4( n+ ). ( u) = ( u ) = () v = p ( v ) = p ( u ) = 6+ ( ); n ( u ) = ( un) + ; n ( v ) = 4 For the vertces v v 3 vn and v v v n we assgn dstnct odd numbers (except p and p ). In vew o the above dened labelng pattern we have e (0) = 4n+ = e (). Thus e (0) e (). Hence D ( Bnn ) s a dvsor cordal graph. Illustraton.8. Dvsor cordal labelng o graph D( B55) s shown n Fgure 4. 74

9 Some Star and Bstar Related Dvsor Cordal Graphs Theorem.9. B nn Fgure 4: Dvsor cordal labelng o graph D( B 55) s a dvsor cordal graph. Proo: Consder B nn wth vertex set { uvu v n} where u v are pendant vertces. Let G be the graph Bnn then VG ( ) = n+ and E( G) = 4n+. We dene vertex labelng : V( G) { n+ } as ollows. Let p be the hghest prme number<n+. ( u) = ( v) = p ( u) = ; ( v ) = 4+ ( ); n For the vertces u u3 un we assgn dstnct odd numbers (except p ). In vew o the above dened labelng pattern we have e (0) = n e () = n+ Thus e (0) e (). Hence B nn s a dvsor cordal graph Illustraton.0. Dvsor cordal labelng o the graph B 77 s shown n Fgure 5. Fgure 5: Dvsor cordal labelng o the graph B 77 75

10 S. K. Vadya and N. H. Shah 3. Concludng Remarks As all the graphs are not dvsor cordal graphs t s very nterestng and challengng as well to nvestgate dvsor cordal labelng or the graph or graph amles whch admt dvsor cordal labelng. Here we have contrbuted some new results by nvestgatng dvsor cordal labelng or some star and bstar related graphs. Varatharajan et al. [9] have proved that Kn and Bnn are dvsor cordal graphs whle we prove that the splttng graphs o star Kn and bstar Bnn are also dvsor cordal graphs. Thus dvsor cordalty remans nvarant or the splttng graphs o K n and B nn. It s also nvarant or degree splttng graph o B nn shadow graph o Bnn and square graph o B nn. Acknowledgement: The authors are hghly thankul to the anonymous reeree or knd comments and constructve suggestons. REFERENCES. J. Gross and J. Yellen Graph Theory and ts applcatons CRC Press (999).. D. M. Burton Elementary Number Theory Brown Publshers Second Edton (990). 3. L. W. Beneke and S. M. Hegde Strongly multplcatve graphs Dscuss. Math. Graph Theory (00) R. L. Graham and N. J. A. Sloane On addtve bases and harmonous graphs SIAM J. Alg. Dsc. Meth. (4) (980) Z. Lang and Z. Ba On the odd harmonous graphs wth applcatons J. Appl. Math. Comput. 9 (009) V. Yegnanaryanan and P. Vadhyanathan Some nterestng applcatons o graph labellngs J. Math. Comput. Sc. (5) (0) J. A. Gallan A dynamc survey o graph labelng The Electronc Journal o Combnatorcs 9 (0) # DS6. Avalable: 8. I. Caht Cordal Graphs A weaker verson o graceul and harmonous Graphs Ars Combnatora 3 (987) M. Andar S. Boxwala and N. B. Lmaye Cordal labelngs o some wheel related graphs J. Combn. Math. Combn. Comput. 4 (00) M. Andar S. Boxwala and N. B. Lmaye A note on cordal labelng o multple shells Trends Math. (00) S. K. Vadya and N. A. Dan Some new star related graphs and ther cordal as well as 3-equtable labelng Journal o Scences () (00) -4.. Nasreen Khan Cordal labellng o cactus graphs Advanced Modelng and Optmzaton 5() (03) J. B. Babujee and L. Shobana Cordal Languages and Cordal Numbers Journal o Appled Computer Scence & Mathematcs 3 (6) (0)

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