THE REVISED EDGE SZEGED INDEX OF BRIDGE GRAPHS

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1 Hacettepe Journal of Mathematics and Statistics Volume THE REVISED EDGE SZEGED INDEX OF BRIDGE GRAPHS Hui Dong and Bo Zhou Received 01:09:011 : Accepted 15:11:011 e=uv EG Abstract The revised edge Szeged index of a connected graph G is defined as SzeG = m m0e G ue G m m0e G ve G where EG is the edge set of G m ue G is the number of edges closer to vertex u than to vertex v in G m ve G is the number of edges closer to vertex v than to vertex u in G and m 0e G is the number of edges equidistant from both ends of e We give a formula for the revised edge Szeged index of a bridge graph from which the revised edge Szeged indices for several classes of graphs are calculated Keywords: Szeged index Revised edge Szeged index Revised Szeged index Bridge graphs Distance 000 AMS Classification: 05C1 05C90 1 Introduction Topological indices are used in theoretical chemistry for the design of chemical compounds with given physicochemical properties or given pharmacologic and biological activities [10] The Wiener index is one of the oldest and the most thoroughly studied topological index Motivated by the original definition of the Wiener index of a tree Gutman [3] introduced the Szeged index which coincides with the Wiener index for a tree It found applications in quantitative structure-property-activity-toxicity modeling see [5] There are some variants of the Szeged index Randić [9] introduced the revised Szeged index which shows to be a better descriptor for structure-property relationships for cyclic molecules Recently the edge Szeged index and the revised edge Szeged index were proposed in [] and [] respectively Department of Mathematics South China Normal University Guangzhou P R China H Dong donghui100@163com B Zhou zhoubo@scnueducn Corresponding author

2 560 H Dong B Zhou Let G be a connected graph with vertex set VG and edge set EG For uv VG d Guv denotes the distance between u and v in G Let e = uv EG w VG The distance between e and w in G is defined as d Gew = min{d Guwd Gvw} Let m ue G be the number of edges closer to vertex u than to vertex v in G and m ve G the number of edges closer to vertex v than to vertex u in G ie m ue G = {f EG : d Gfu < d Gfv} m ve G = {f EG : d Gfv < d Gfu} The edge Szeged index of G is defined as [] Sz eg = e=uv EG m ue Gm ve G Some basic properties of the edge Szeged index have been established see [1 6 11] Let m 0e G be the number of edges equidistant from both ends of e = uv EG ie m 0e G = {f EG : d Gfu = d Gfv} The revised edge Szeged index of G is defined in [] as Sz eg = e=uv EG m m0e G ue G m m0e G ve G Let {G i} d be a set of finite pairwise vertex-disjoint connected graphs with v i VG i The bridge graph BG 1G G d = BG 1G G d ; v 1v v d of {G i} d with respect to the vertices {v i} d is the graph obtained from the graphs G 1G G d by connecting the vertices v i and v i1 by an edge for all i = 1 Formulae for the Szeged index edge Szeged index revised Szeged index of bridge graphs have been given in [7 13 8] respectively Here we give a formula for the revised edge Szeged index of a bridge graph from which the revised edge Szeged indices for several classes of graphs are calculated Results For a connected graph G with w VG let L Gw and R Gw be respectively the sets of edges e = uv in EG such that d Guw < d Gvw and d Guw > d Gvw and Q Gw the set of edges e = uv in EG such that d Guw = d Gvw To make this well-defined we choose an arbitrary direction on the edges of G which is fixed for all the following computations; the results do not depend on the direction chosen In a bridge graph BG 1G G d for i = 1d let L i = L Gi v i R i = R Gi v i and Q i = Q Gi v i

3 The Revised Edge Szeged Index of Bridge Graphs Lemma [13] The edge Szeged index of the bridge graph G = BG 1G G d of {G i} d with respect to the vertices {v i} d is given by Sz eg = d Sz eg i d EG EG i m ve G i e=uv L i α i EG α i 1 m ue G i e=uv R i where α i = i j=1 EGj i 1 for all i = 1d Theorem The revised edge Szeged index of the bridge graph G = BG 1G G d of {G i} d with respect to the vertices {v i} d is given by Sz eg = d SzeG 1 i d EG EG i Q i d EG EG i l i r i α i EG α i 1 1 EG 1 where l i = e=uv L i r i = α i = e=uv R i m ve G i 1 m0e Gi m ue G i 1 m0e Gi i EG j i 1 j=1 for all i = 1d Proof Let EG = m and EG i = m i Obviously m ue Gm ve Gm 0e G = m and m ue G i m ve G i m 0e G i = m i Note that EG i = Q i L i R i for i = 1d From the definition of the revised edge Szeged index we have SzeG = m ue Gm ve G e=uv EG [ ] m0e G m ue Gm ve G m 0e G e=uv EG [ ] m0e G = Sz eg m m 0e G m 0e G e=uv EG = Sz 1 eg m 0e G m m0e G e=uv EG

4 56 H Dong B Zhou = Sz eg 1 1 d d e=uv Q i m 0e G e=uv L i R i m 0e G m m0e G m m0e G 1 m 0v iv i1 G m m0vivi1 G For i = 1d if e = uv Q i then all the edges in EG\EG i are equidistant from both ends of the edge e = uv and thus m 0e G = m 0e G im m i Then d m 0e G m m0e G e=uv Q i d = m 0e G im m i m m0e Gim mi e=uv Q i d = m 0e G i m i m0e Gi e=uv Q i 1 d m m i e=uv Q i d = m 0e G i m i m0e Gi e=uv Q i 1 d m m i Q i For i = 1d if e = uv L i R i then there is no edge in EG\EG i which is equidistant from both ends of the edge e = uv and thus m 0e G = m 0e G i Then d m 0e G m m0e G e=uv L i R i d = m 0e G i m m0e Gi e=uv L i R i d = m 0e G i m i m0e Gi e=uv L i R i d m 0e G im m i e=uv L i R i d = m 0e G i m i m0e Gi e=uv L i R i d m m i m 0e G i e=uv L i R i

5 The Revised Edge Szeged Index of Bridge Graphs 563 For i = 1d it is obvious that m 0v iv i1 G = 1 Thus m 0v iv i1 G m m0vivi1 G = It follows that By Lemma 1 Sz eg = Sz eg 1 Sz eg = 1 d Sz eg i d e=uv EG i d m m i Q 1 i 1 m 1 m 1 m 0e G i m i m0e Gi d m m i d m m i m ve G i e=uv L i α im α i d d m m i Q i 1 m 1 e=uv EG i e=uv L i R i m 0e G i m 0e G i m ue G i e=uv R i m i m0e Gi d m 0e G i m m i e=uv L i R i as desired = d SzeG 1 i d m m i Q i d m m il i r i α im α i 1 1 m 1 For a connected graph H with vertex w let G d Hw be the bridge graph BG 1G G d = BG 1G G d ;v 1v v d with G 1 = G = = G d = H and v 1 = v = = v d = w ie G d Hw = BHH;ww }{{}}{{} dtimes dtimes 3 Corollary Let H be a connected graph with m edges Then the revised edge Szeged index of the bridge graph G d Hw is given by Sz eg d Hw = dsz eh 1 dm1dmdm 1 QHw dm1l H r H m1dmdmd 1

6 56 H Dong B Zhou where l H = r H = e=uv L Hw e=uv R Hw m ve H 1 m0e H m ue H 1 m0e H Proof Since EH = m we have EG d Hw = dm and α i = i j=1 EH i 1 = m1i 1 Thus α i EG d Hw α i 1 = [m1i 1][dm m1i 1 1] = [ m1 i m1 di m1d1] m1dmdmd 5 = 6 By Theorem 1 it is easily seen that Sz eg d Hw = d SzeH 1 1 d EG d Hw EH Q Hw d EG d Hw EH l H r H α i EG d Hw α i 1 = dsz eh 1 1 EG d Hw 1 d dm m Q Hw d dm ml H r H m1dmdmd dm 1 1 = dsz eh 1 dm1dmdm 1 QHw as desired dm1l H r H m1dmdmd 1

7 The Revised Edge Szeged Index of Bridge Graphs 565 Let P n = u 1u u n be the path on n vertices Obviously P n = G np 1u 1 By Corollary 3 SzeP nn n = n 1 1 n = 1 3 n 1 Corollary The revised edge Szeged index of the bridge graph A dn = G d P nu 1 is given by SzeA n d n dn = d3d n d nd 3nd1 Proof Note that SzeP n = n 3 l Pn r Pn = By Corollary 3 we have as desired Sz ea dn e=uv L Pnu1 1 n 1 and m 1 ve P n n 1 = 1 n 1 e=uv R Pnu1 m 1 ue P n = dszep 1 n dndn 1dn ndn 1dn 1d dnl Pn r Pn 1 [ ] [ ] n = d 1 3 n 1 n 1 dn 1 n 1 [ n dd1 1 ] n [ ] [ n = d 1 3 n 1 d n 3 3 n d1 1 3 n n d n = d3d n d1 3 3 Let C n = u 1u u nu 1 be the cycle on n vertices n ] 1 nd 3nd1 5 Corollary The revised edge Szeged index of the bridge graph T dn = G d C nu 1 is given by 1 dn3 1 dn1dndn n 1 n1dndnd 1 if n is odd SzeT dn = 1 dn3 1 dn n1 n1dndnd 1 if n is even

8 566 H Dong B Zhou Proof By direct calculation we have SzeC n = n3 If n is odd then QCnu 1 = 1 and l Cn r Cn = m 1 ve C n m 1 ue C n e=uv L Cnu1 = 1 nn 1 By Corollary 3 we have e=uv R Cnu1 Sz et dn = d n3 1 dn1dndn 1 1 dn1 1 nn 1 n1dndnd 1 = 1 dn3 1 dn1dndn n 1 n1dndnd 1 If n is even then Q Cnu1 = 0 l Cn r Cn = n and arguing as above we have the desired expression for SzeT dn Acknowledgement This work was supported by the National Natural Science Foundation of China Grant No References [1] Cai X and Zhou B Edge Szeged index of unicyclic graphs MATCH Commun Math Comput Chem [] Dong H Zhou B and Trinajstić N A novel version of the edge-szeged index Croat Chem Acta [3] Gutman I A formula for the Wiener number of trees and its extension to graphs containing cycles Graph Theory Notes N Y [] Gutman I and Ashrafi AR The edge version of the Szeged index Croat Chem Acta [5] Khadikar P V Karmarkar S Agrawal V K Singh J Shrivastava A Lukovits I and Diudea M V Szeged index Applications for drug modeling Lett Drug Design Disc [6] Khalifeh MH Yousefi-Azari H Ashrafi AR and Gutman I The edge Szeged index of product graphs Croat Chem Acta [7] Mansour T and Schork M The vertex PI index and Szeged index of bridge graphs Discrete Appl Math [8] Mansour T and Trinajstić N The revised Szeged index of bridge graphs to appear [9] Randić M On generalization of Wiener index to cyclic structures Acta Chim Slov [10] Trinajstić N Chemical Graph Theory nd edn CRC Press Boca Raton 199 [11] Vukičević D Note on the graphs with the greatest edge-szeged index MATCH Commun Math Comput Chem [1] Wiener H Structural determination of paraffin boiling points J Am Chem Soc [13] Xing R and Zhou B On the revised Szeged index of bridge graphs Comptes Rendus Math