Bounds for the Connective Eccentric Index

Transcription

1 It. J. Cotemp. Math. Sceces, Vol. 7, 0, o. 44, 6-66 Bouds for the Coectve Eccetrc Idex Nlaja De Departmet of Basc Scece, Humates ad Socal Scece (Mathematcs Calcutta Isttute of Egeerg ad Maagemet Kolkata, West Begal, Ida Abstract ξ deg( v The coectve eccetrc dex of a graph G s defed as C ( G = v V ( G ε ( v where ε ( v ad deg(v deote the eccetrcty ad degree of the vertex v respectvely. I ths paper we derve some ew upper ad lower bouds for the coectve eccetrc dex terms of some other graph varats such as maxmum ad mmum degree, radus, dameter, frst Zagreb dex ad frst Zagreb eccetrcty dex. Keywords: Eccetrcty, Degree, Graph Ivarat, Coectve Eccetrc Idex. Itroducto Let G be a smple coected graph wth vertex set V(G ad edge set E(G so that V ( G = ad E( G = m. For ay vertex v V ( G, let deg(v deote the umber of frst eghbor.e. degree of the vertex v. Also let δ = δ ( G ad = ( G deote the mmum ad maxmum degree of all the vertces of G, respectvely. For vertces u ad v V ( G, let the dstace betwee u ad v deoted by d(u,v whch s defed as legth of the shortest path coectg u ad v. The eccetrcty of a vertex v V ( G, deoted by ε ( v, s the dstace betwee v ad a vertex farthest from v.e. ε ( v = max{ d( v, x : x V ( G}. The radus r = r( G ad dameter d = d ( G of a graph s the mmum ad maxmum eccetrcty amog the vertces of G.e. r = r( G = m{ ε ( v : v V ( G} ad d = d( G = max{ ε ( v : v V ( G} respectvely. Also the total eccetrcty of a graph, deoted by θ ( G, s the sum of all the eccetrctes of G [5, 8]. The classcal Zagreb dces, troduced by Gutma [4], are the oldest ad most popular graph varat, defed as the sum of squares of degrees of the vertces ad sum of product of the degrees of the adjacet vertces of a graph. Aalogous

2 6 Nlaja De to Zagreb dces aother two topologcal dces were troduced by M. Ghorba et al. [6] ad D. Vukcevc et al. [] ad defed by replacg degrees by eccetrcty of the vertces, whch are termed as frst ad secod Zagreb eccetrcty dces ad deoted by E ( G ad E ( G. I recet years a umber of graph varats related to eccetrcty have bee derved ad studed such as eccetrc coectvty dex, augmeted eccetrc coectvty dex, eccetrc dstace sum dex ad so o. The coectve eccetrc dex s also of ths type ad s relatvely ew ad ot yet gets much atteto. Ths dex was troduced by Gupta, Sgh ad Mada [9] ad s defed as ξ deg( v C ( G = v V ( G ε ( v M. Ghorba [5] preseted some bouds of coectve eccetrc dex ad also compute ths dex for two fte class of fullerees. The problem of determg extremal propertes ad the correspodg extremal graphs of some graph varats were subject to a large umber of studes [,3,7]. I ths paper we vestgate some ew upper ad lower bouds of coectve eccetrc dex terms of umber of vertces (, umber of edges (m, maxmum vertex degree (, mmum vertex degree (δ, radus (r, dameter (d, total eccetrcty( θ ( G, the frst Zagreb dces (M (G ad the frst Zagreb eccetrcty dex ( E ( G. Ma Results Theorem Let G be a smple coected graph wth radus r ad dameter d, the m ξ m C ( G d r wth equalty f ad oly f all the vertces of G are of same eccetrcty. Proof: Sce we have for ay v V ( G, r ε ( v d ad deg( v = m, from v V ( G the defto of coectve eccetrc dex we get the desred result. Obvously ths equalty equalty holds f ad oly f ε ( v = r = d for all v V ( G. Corollary Let G be a smple coected graph wth 4 vertces for whch ts complemetg s also coected the ξ ξ ( C ( G + C ( G wth equalty f ad oly f all the vertces of G ad G are of eccetrcty. Proof: Sce, both G ad G coected graph, each has radus at least two. Also

3 Bouds for the coectve eccetrc dex 63 ( sce the total umber of edges of G ad G s, so by Theorem we have the desred result wth equalty f ad oly f all the vertces of G ad G are of eccetrcty. Theorem Let G be a smple coected graph wth ( 3 vertces, the ξ δ C ( G δ wth equalty f ad oly f all the vertces of G are of same degree ad eccetrcty ad deg( v + ε ( v =. Proof: We have for ay v V ( G, ε ( v deg( v, wth equalty acheved for G K je for j =,,..., or G P 4. So from the defto of coectve eccetrc dex ξ deg( v deg( v C ( G = ( ε ( v deg( v v V ( G v V ( G Now, sce for ay v V ( G, deg( v δ, wth equalty f ad oly f G s deg( v δ regular graph, we have for ay v V ( G,, from where we get deg( v δ the desred result. The equalty holds ( whe G K je for j =,,..., or G P4. Amog these graphs oly for regular graphs the lowed boud of ths theorem attas. So, ths theorem equalty holds whe all the vertces of G are of same degree ad hece eccetrcty so that deg( v + ε ( v = for ay v V ( G. Theorem 3 Let G be a smple coected graph the ( ξ M G C ( G r wth equalty f ad oly f all the vertces of G are of same degree ad eccetrcty. Proof: From defto of coectve eccetrc dex of G, we have Sce, v V ( G ξ deg( v C ( G = deg( v ( ε ( v ε ( v ε ( v v V ( G ε v v V ( G v V ( G v V ( G ad ε ( v r for all v V ( G, wth equalty f ad oly f all the vertces are of same eccetrcty, the desred result follows from above. Obvously ths theorem equalty holds f ad oly f all the vertces are of same degree ad eccetrcty.

4 64 Nlaja De Theorem 4 Let G be a smple coected graph o vertces. Let 0 be the umber of vertces wth eccetrcty oe G, the ξ 0 C ( G ( wth equalty f ad oly f G K, where ( 0 s eve. Proof: Sce 0 be the umber of vertces wth eccetrcty oe G, so the remag ( 0 vertces are of eccetrcty at least two. Let S = { v, v, v3,..., v } be the set of vertces such that ε ( v 0 = for =,,..., 0. Also t follows that the vertces u V ( G \ S, ε ( u ad deg( u (. The from the defto of coectve eccetrc dex, we have 0 0 ξ deg( v deg( u C ( G = + deg( v + ( ( 0 ε ( v ε ( u = u V ( G\ S = Now sce 0 deg( v 0 (, the desred result follows from above. Clearly, = 0 ths theorem equalty holds f ad oly f G K, where ( 0 s eve. Theorem 5 Let G be a smple coected graph o vertces ad m edges, the ξ m C ( G θ ( G I the above equalty equalty holds f ad oly f all the vertces of G are of same eccetrcty. Proof: We wll prove ths theorem usg the followg Chebyschev s equalty: Let a ad b are real umbers, the ( a b a b = = = wth equalty holds f ad oly f a = a =... = a or b = b =... = b. Now settg a = deg( v ad b =, for =,,...,, we get from ( ε ( v ξ deg( v C ( G = deg( v = m = ε ( v = = ε ( v = ε ( v Now usg the equalty betwee arthmetc mea ad harmoc mea, we have = = ε ( v θ ( G ε ( v (3 =

5 Bouds for the coectve eccetrc dex 65 wth equalty holdg f ad oly f ε ( v = ε ( v =... = ε ( v. Thus from (3 we get the desred result. Obvously ths theorem equalty holds f ad oly f all the vertces of G are of same eccetrcty, for example, f ad oly f G K or G C or G K m, or the graph G obtaed fromg K by removg a perfect matchg. ξ Note that ths s a mproved verso of the lower boud of C ( G gve [5]. Theorem 6 Let G be a smple coected graph, the ξ δ rd C ( G M( G + rδ + d E ( G wth equalty holdg f ad oly f all the vertces of G are of same degree ad eccetrcty. Proof: We use the followg Daz-Metcalf equalty to prove ths theorem: If a ad b, for =,,..., are real umbers such that ha b Hb, for =,,...,, the b + hh a ( H + h ab = = = (4 I the above equalty equalty holds f ad oly f b = ha or b = Ha for every =,,...,. By settg b = deg( v ad a = for every ε ( v =,,...,, we have from (4 deg( v deg( v + hh ( H + h = = ε ( v = ε ( v Hece usg defto of coectve eccetrc dex, frst Zagreb dex ad frst Zagreb eccetrcty dex ad also cosderg the equalty betwee arthmetc mea ad harmoc mea, we have from above C ξ ( G M( G + hh (5 H + h E ( G Now sce, δ deg( v ad r ε ( v d for =,,...,, so we have h = δr ad H = d. Thus from (5 we get the desred result. δ r I the above equalty equalty holds f ad oly f deg( v = or ε ( v d deg( v = for =,,...,.e. f ad oly f all the vertces are of same ε ( v degree ad eccetrcty.

6 66 Nlaja De Refereces [] A. Ilć, M. Ilć, B. Lu, O the upper bouds for the frst Zagreb Idex, Kragujevac J. Math., 35 (0, [] D. Vukčevć ad A. Graovac, Note o the comparso of the frst ad secod ormalzed Zagreb eccetrcty dces, Acta Chm. Slov., 57 (00, [3] H. Hua, G.Yu, Bouds for the adjacet eccetrc dstace sum, It. Math. Forum, 7 (0, [4] I. Gutma, N. Trajstć, Graph theory ad molecular orbtals, Total π-electro eergy of alterat hydrocarbos, Chem. Phys. Lett., 7 (97, [5] M. Ghorba, Coectve Eccetrc Idex of Fullerees, J. Math. Naoscece, (0, [6] M. Ghorba ad M. A. Hossezadeh, A ew verso of Zagreb dces, Flomat, 6 (0, [7] N. De, Some bouds of reformulated Zagreb dces, Appl. Math. Sc., 6 (0, [8] P. Dakelma, W. Goddard, C.S. Swart, The average eccetrcty of a graph ad ts subgraphs, Utl. Math., 65 (004, 4-5. [9] S. Gupta, M. Sgh, A. K. Mada, Coectve eccetrcty Idex: A ovel topologcal descrptor for predctg bologcal actvty, J. Mol. Graph. Model., 8 (000, 8-5. Receved: July, 0