Research Article On Some Bounds and Exact Formulae for Connective Eccentric Indices of Graphs under Some Graph Operations

Transcription

1 Iteratioal Combiatorics, Article ID 57957, 5 pages Research Article O Some Bous a Exact Formulae for Coective Eccetric Iices of Graphs uer Some Graph Operatios Nilaja De, Aita Pal, a Sk M Abu Nayeem 3 Departmet of Basic Scieces a Humaities (Mathematics), Calcutta Istitute of Egieerig a Maagemet, Kolkata , Iia Departmet of Mathematics, Natioal Istitute of Techology, Durgapur 73 09, Iia 3 DepartmetofMathematics,AliahUiversity,DN0,SectorV,SaltLake,Kolkata70009,Iia Correspoece shoul be aresse to Sk M Abu Nayeem; ayeemmath@aliahaci Receive 4 July 04; Accepte 4 November 04; Publishe 4 December 04 Acaemic Eitor: Cai Heg Li Copyright 04 Nilaja De et al This is a ope access article istribute uer the Creative Commos Attributio Licese, which permits urestricte use, istributio, a reprouctio i ay meium, provie the origial work is properly cite The coective eccetric iex of a graph is a topological iex ivolvig egrees a eccetricities of vertices of the graph I this paper, we have stuie the coective eccetric iex for ouble graph a ouble cover Also we give the coective eccetric iex for some graph operatios such as jois, symmetric ifferece, isjuctio, a splice of graphs Itrouctio Let G beasimplecoectegraphwithvertexsetv(g) a ege set E(G) Let a m be the umber of vertices a eges of G, respectively We eote the egree of a vertex V of G by G (V) Foru, V V(G), theistacebetweeu a V i G is efie as the legth of the shortest path betwee u a V i G a is eote by G (u, V) For a give vertex V of G, the eccetricity ε G (V) is the largest istace from V to ay other vertices of G The sum of eccetricities of all the vertices of G is eote by θ(g) If ay vertex V V(G) is ajacet to all the other vertices of G the V is calle a well-coecte vertex Thus, if V V(G) is a well-coecte vertex, the ε G (V) = For example, all the vertices of a complete graph are well coecte Recetly, a umber of topological iices ivolvig vertex egree a eccetricity were subject to a lot of mathematical as well as chemical stuies A topological iex of this type, itrouce by Gupta et al, was ame as the coective eccetric iex a was efie as C ξ (G) = G (V) ε G (V) () V V(G) Ghorbai 3 gave some bous of coective eccetricity iex a also compute this iex for two ifiite classes of erimers De 4 reporte some bous for this iex i terms of some graph ivariats such as maximum a miimum egree, raius, iameter, first Zagreb iex, a first Zagreb eccetricity iex I 5, Ghorbai a Malekjai compute the eccetric coectivity iex a the coective eccetric iex of a ifiite family of fullerees I 6, Yu a Feg also erive some upper or lower bous for the coective eccetric iex a ivestigate the maximal a the miimal values of coective eccetricity iex amog all -vertex graphs with fixe umber of peet vertices I 3, Ghorbai showe that, for a vertex trasitive graph G, the coective eccetric iex is give by C ξ (G) = t A G (x i ) j j= ε G (x i ), () where A,A,,A t are the orbits of Aut(G) uer its atural actio o V(G) a x i A i, i ti particular, if G is a regular graph, the C ξ (G) = δ/r(g), where istheumberofverticesofg, whichisaδ-regular graph, a r(g) is the raius of G LetK, C, Q m, Π m, A m eote the complete graph with vertices, the cycle o vertices, m-imesioal hypercube, m-sie prism, a the m-sie atiprism, respectively It ca be easily verifie that

2 Iteratioal Combiatorics the explicit formulae for the coective eccetric iex of K, C, Q m, Π m, A m are as follows x x x 3 Propositio Cosier the followig Propositio Cosier the followig C ξ (K )=( ) (3) C ξ (C )= / (4) Propositio 3 Cosier the followig Propositio 4 Cosier the followig m C ξ { m+, (Π m )= { m { m+, Propositio 5 Cosier the followig C ξ (Q m )= m (5) whe m is eve, whe m is o (6) C ξ (A m )= { 6, whe m is eve, { 6m { m+, whe m is o (7) Several stuies o ifferet topological iices relate to graph operatios of ifferet kis are available i the literature 7 I this paper, first we calculate coective eccetric iex of ouble graph a ouble cover a hece the explicit formulae for the coective eccetric iices of joi, symmetric ifferece, isjuctio, a splice of graphs are obtaie For the efiitios a ifferet results o graph operatios, such as joi, symmetric ifferece, a isjuctio, reaers are referre to the book of Imrich a Klavžar 3 y y y 3 Figure : The graph P 3 a its ouble graph P 3 Proof From the costructio of ouble graph, it is clear that G (x i )= G (y i )=,wherev i V(G) a x i,y i V(G ) are the correspoig cloe vertices of V i Alsowe ca write ε G (x i )=ε G (y i )=ε G (V i ) whe ε G (V i ) a ε G (x i )=ε G (y i )=ε G (V i )+whe Thus the coective eccetric iex of ouble graph G is C ξ (G )= G (x i ) ε G (x i ) + G (y i ) ε G (y i ) = =4 =4 ε G (V i ) ε G (V i ) ε G (V i ) ε G(V i)= ε G(V i)= =4C ξ (G) ( ) G ε G (V i )+ (8) Mai Results I this sectio, first we efie a the compute eccetric coectivity iex of ouble graph a ouble cover graph Coective Eccetric Iex of Double Graph a Double Cover Let us eote the ouble graph of a graph G by G, which is costructe from two copies of G i the followig maer 3, 4 Let the vertex set of G be V(G) = {V, V,,V },atheverticesofg aregivebythetwosets X={x,x,,x } a Y={y,y,,y }Thus,foreach vertex V i V(G), therearetwoverticesx i a y i i V(G ) The ouble graph G icluestheiitialegesetofeachcopy of G,a,forayegeV i V j E(G),twomoreegesx i y j a x j y i are ae The graph P 3 a its ouble graph P 3 is show i Figure Theorem 6 The coective eccetric iex of the ouble graph G is give by C ξ (G )=4C ξ (G) ( ) G,where G is the umber of vertices with egree,thatis,of eccetricity oe Let G = (V, E) be a simple coecte graph with V = {V, V,,V } The extee ouble cover of G, eote by G, is the bipartite graph with bipartitio (X, Y) where X= {x,x,,x } a Y={y,y,,y } i which x i a y j are ajacet if a oly if either V i a V j are ajacet i G or i=j For example, the extee ouble cover of the complete graph is the complete bipartite graph This costructio of the extee ouble cover was itrouce by Alo 3 i 986 Extee ouble cover of P 3 is illustrate i Figure Theorem 7 The coective eccetric iex of the extee ouble cover G satisfies the iequality C ξ (G )<C ξ (G) ( G ) (9) Proof If G is a graph with vertices a m eges, the, from efiitio, the extee ouble cover graph G cosists of vertices a ( + m) eges a G (x i ) = G (y i ) = +a ε G (x i )=ε G (y i )=ε G (V i )+for,,,

3 Iteratioal Combiatorics 3 3 x x x 3 y y y 3 Figure : The graph P 3 a its extee ouble cover P 3 Thus the coective eccetric iex of extee ouble cover graph G is give by C ξ (G )= G (x i ) ε G (x i ) + G (y i ) ε G (y i ) = + ε G (V i )+ + + < G (V i )+ + { ε G (V i ) G (V i )+} = + = + + ε G (V i ) ε G (V i ) + =C ξ (G) + { G } ( ) G + G Proof Let a be the umbers of vertices i G a G, respectively Thus, V(G +G ) = V + V = + For vertices u V a V V,itholsthat G +G (u) = G (u) + a G +G (V) = G (V) + Sice oe of G a G cotais well-coecte vertices, the, for every u V(G +G ), ε G +G (u) = So, from efiitio of coective eccetric iex, we have C ξ (G +G ) = V V(G ) = G +G (V) ε G +G (V) + V V(G ) V V(G ) { G (V) + }+ G +G (V) ε G +G (V) V V(G ) { G (V) + } () Now we geeralize the above result for isjoit graphs G,G,,G Theorem 9 Let G,G,,G be graphs with isjoit vertex sets V i =V(G i ) a ege sets E i =E(G i ), i,without well-coecte vertices The, C ξ (G +G + +G )= E(G i) + { V(G i) } V(G i) () Proof From efiitio of joi, we have G +G + +G (V j ) = Gi (V j )+ V(G +G + +G ) V(G i ) a sice oe of G,G,,G cotais ay well-coecte vertex, we have ε G +G + +G (V j )=for all V j V(G i ) Thus the coective eccetric iex of G +G + +G is give by C ξ (G +G + +G ) =C ξ (G) { G } (0) = V(G i ) j= G +G + +G (V j ) ε G +G + +G (V j ) Now some exact formulae for the eccetric coectivity iex of jois, symmetric ifferece, isjuctio, a splice graphs are presete = V(G i ) j= Gi (V j )+ V(G +G + +G ) V(G i) Joi The joi G +G of two graphs G a G,with isjoit vertex sets V, V a ege sets E, E,isthegraph uio G G together with all the eges joiig V a V ; that is, G +G cosists of the vertex set V V a ege set E E {xy:x V,y V } Theorem 8 Let G a G be two graphs without wellcoecte vertices The, C ξ (G +G )= E(G ) + E(G ) + V(G ) V(G ) = E(G ) + E(G ) + + E(G ) + V(G +G + +G ) { V(G ) + V(G ) + + V(G ) } { V(G ) + V(G ) + + V(G ) } (3)

4 4 Iteratioal Combiatorics m m V V Proof Sice the istace betwee ay two vertices of a symmetric ifferece caot excee two, if oe of the compoets cotais well-coecte vertices, the eccetricity of all vertices is costat a is equal to two; that is, ε G G (u,u )=,forallvertices(u,u ) 7, 8 Thus the coective eccetric iex of symmetric ifferece G G of two graphs G a G is give by C ξ (G G ) m V = (u,u ) V(G G ) G G (u,u ) ε G G (u,u ) Figure 3: The complete -partite graph The followig corollaries are irect cosequeces of the theorem Corollary 0 If G eotes the joi of copies of G,the C ξ (G) = E (G) + ( ) V (G) (4) Corollary Let G=K m,m,,m be the complete -partite graph see Figure 3havigV = V(G) umber of vertices Here the vertex set ca be partitioe ito subsets V,V,,V such that V V V = V(G) a V i V j =φ, i =j, i, j FromefiitioofjoiitisclearthatG=K m,m,,m is the joi of empty graphs with m,m,,m umber of vertices The the coective eccetric iex of G=K m,m,,m is give by C ξ (K m,m,,m )= i j i=j m i m j (5) 3 Symmetric Differece Let G a G be two graphs with vertex sets V(G ) a V(G ) a ege sets E(G ) a E(G ) The the symmetric ifferece of G a G, eote by G G,isthegraphwithvertexsetV(G ) V(G ) i which ay two vertices (u,u ) are ajacet to (V, V ) wheever u is ajacet to V i G or u is ajacet to V i G,butot both From efiitio of symmetric ifferece, the egree of avertex(u,u ) of G G is give by G G (u,u )= V(G ) G (u )+ V(G ) G (u ) G (u ) G (u ) (6) Theorem The coective eccetric iex of the symmetric ifferece G G of two graphs G a G is give by C ξ (G G )= E(G ) V(G ) + E(G ) V(G ) 4 E(G ) E(G ), (7) where oe of G a G cotais well-coecte vertices = u V(G ) u V(G ) = V(G ) u V(G ) u V(G ) V(G ) G (u )+ V(G ) G (u ) G (u ) G (u ) G (u ) G (u )+ V(G ) u V(G ) G (u ) u V(G ) G (u ) (8) 4 Disjuctio The isjuctio G G of two graphs G a G is the graph with vertex set V(G ) V(G ) i which (u,u ) is ajacet to (V, V ) wheever u is ajacet to V i G or u is ajacet to V i G Obviously,theegreeofa vertex (u,u ) of G G is give by 7, 8 (G G ) (u,u )= V(G ) G (u ) + V(G ) G (V ) G (u ) G (V ) (9) Theorem 3 The coective eccetric iex of the isjuctio G G of two graphs G a G is give by C ξ (G G )= E(G ) V(G ) + E(G ) V(G ) E(G ) E(G ), (0) whereoeofg a G cotais well-coecte vertices Proof Sice the istace betwee ay two vertices of a isjuctio caot excee two, if oe of the compoets cotais well-coecte vertices, the eccetricity of all vertices is costata equal to two 7, 8 The the coective

5 Iteratioal Combiatorics 5 eccetric iex of the isjuctio G G of two graphs G a G is compute as C ξ (G G ) = = (u i,v j ) V(G G ) u V(G ) V V(G ) G G (u,u ) ε G G (u,u ) V(G ) G (u )+ V(G ) G (u ) G (u ) G (u ) () 5 Splice Let G = (V,E ) a G = (V,E ) be two graphs V V = φletv V a V V be two give vertices of G a G,respectivelyAspliceofG a G at the vertices V a V is obtaie by ietifyig the vertices V a V i the uio of G a G a is eote by S(G,G, V, V ) 5 Differet topological iices of the splice graphs have alreay bee compute 6 Let u V(S); the ε (u) eotes the eccetricity of u as a vertex of G,ε (u) eotes the eccetricity of u as a vertex of G,aε(u) eotes the eccetricity of u as a vertex of SLet S (x) be the egree of the vertex x V(S) The the coective eccetric iex of S(G,G, V, V ) is compute as follows Theorem 4 The coective eccetric iex of splice of G a G is give by C ξ (S) = S (x) x V max { G (x, V )+ε (x) ε (V )} () + S (y) max { G (y, V )+ε (V )ε (y)} y V Proof For ay vertex x V, Sharafii a Gutma 6 showe that ε (x) = max { G (x, V )+ε (x) ε (V )} (3) S Gupta, M Sigh, a A K Maa, Coective eccetricity iex: a ovel topological escriptor for preictig biological activity, Molecular Graphics a Moellig, vol8, o, pp 8 5, M Ghorbai, Coective eccetric iex of fullerees, Joural of Mahematical Naosciece, vol, pp 43 5, 0 4 N De, Bous for the coective eccetric iex, Iteratioal Cotemporary Mathematical Scieces,vol7,o 4 44, pp 6 66, 0 5 M Ghorbai a K Malekjai, A ew metho for computig the eccetric coectivity iex of fullerees, Serica Joural of Computig,vol6,o3,pp99 308,0 6 G Yu a L Feg, O coective eccetricity iex of graphs, MATCH: Commuicatios i Mathematical a i Computer Chemistry,vol69,o3,pp6 68,03 7 A R Ashrafi, M Ghorbai, a M A Hossei-Zaeh, The eccetric coectivity polyomial of some graph operatios, Serica Computig,vol5,o,pp0 6,0 8 M Ghorbai a M A Hosseizaeh, A ew versio of Zagreb iices, Filomat, vol 6, o, pp 93 00, 0 9 S Hossei-Zaeh, A Hamzeh, a A R Ashrafi, Wier-type ivariats of some graph operatios, Filomat, vol 3, o 3, pp 03 3, M H Khalifeh, H Yousefi-Azari, a A R Ashrafi, The hyper- Wieer iex of graph operatios, Computers & Mathematics with Applicatios,vol56,o5,pp40 407,008 M H Khalifeh, H Yousefi-Azari, a A R Ashrafi, The first a seco Zagreb iices of some graph operatios, Discrete Applie Mathematics,vol57,o4,pp804 8,009 W Imrich a S Klavžar, Prouct Graphs: Structure a Recogitio, Joh Wiley & Sos, New York, NY, USA, N Alo, Eigevalues a expaers, Combiatorica, vol6, o, pp 83 96, H Hua, S Zhag, a K Xu, Further results o the eccetric istace sum, Discrete Applie Mathematics, vol 60, o -, pp70 80,0 5 T Došlic, Splices, liks a their egree-weighte Wieer polyomials, Graph Theory Notes, vol 48, pp 47 55, R Sharafii a I Gutma, Splice graphs a their topological iices, Kragujevac Sciece, vol 35, pp 89 98, 03 Similarly, for ay vertex y V, ε(y)=max { G (y, V )+ε (V )ε (y)} (4) Thus, the esire result follows from the efiitio of coective eccetric iex Coflict of Iterests The authors eclare that there is o coflict of iterests regarig the publicatio of this paper Refereces P Dakelma, W Goar, a C S Swart, The average eccetricity of a graph a its subgraphs, Utilitas Mathematica, vol 65, pp 4 5, 004

6 Avaces i Operatios Research Avaces i Decisio Scieces Applie Mathematics Algebra Probability a Statistics The Scietific Worl Joural Iteratioal Differetial Equatios Submit your mauscripts at Iteratioal Avaces i Combiatorics Mathematical Physics Complex Aalysis Iteratioal Mathematics a Mathematical Scieces Mathematical Problems i Egieerig Mathematics Discrete Mathematics Discrete Dyamics i Nature a Society Fuctio Spaces Abstract a Applie Aalysis Iteratioal Stochastic Aalysis Optimizatio