**Authors**

Arak University

**Abstract**

Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is a

graph $\Gamma_G$ as follows: Take $G\setminus Z(G)$ as vertices of

$\Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever

$yx\neq yx$. $\Gamma_G$ is called the non-commuting graph of $G$.

In recent years many interesting works have been done in non-commutative graph of groups.

Computing the clique number, chromatic number, Szeged index and Wiener index play important role in graph theory. In particular, the clique

number of non-commuting graph of some the general linear groups has been determined.

\nt Recently, Wiener and Szeged indices

have been computed for $\Gamma_{PSL(2,q)}$, where $q\equiv 0 (mod

~~4)$. In this paper we will compute the Szeged index for

$\Gamma_{PSL(2,q)}$, where $q\not\equiv 0 (mod ~~ 4)$.

graph $\Gamma_G$ as follows: Take $G\setminus Z(G)$ as vertices of

$\Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever

$yx\neq yx$. $\Gamma_G$ is called the non-commuting graph of $G$.

In recent years many interesting works have been done in non-commutative graph of groups.

Computing the clique number, chromatic number, Szeged index and Wiener index play important role in graph theory. In particular, the clique

number of non-commuting graph of some the general linear groups has been determined.

\nt Recently, Wiener and Szeged indices

have been computed for $\Gamma_{PSL(2,q)}$, where $q\equiv 0 (mod

~~4)$. In this paper we will compute the Szeged index for

$\Gamma_{PSL(2,q)}$, where $q\not\equiv 0 (mod ~~ 4)$.

**Keywords**

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groups and an application to their non-commuting graphs, J. Algebr. Comb. 34, 638-710, (2011).

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[6] D.M. Cvetkocic, M. Doob, H. Sachs, Spectar of graph theory and application, Academic Press, New York, 1980.

[7] P. Dankelmann, Average distance and independence numbers, Discrete Appl. Math., 51, 75-83, (1994).

[8] A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math.,

66, 211-249, (2001).

[9] A.A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publications De

L‘Institut math´ematiques Nouvelle s´erie, 56, 18-22, (1994).

[10] I. Gutman, A formula for the Wiener number of trees and its extension to graphs cycles, Graph theory Notes of

New York, 27, 9-15, (1994).

[11] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

[12] B. Huppert, and N. Blackburn, Finite groups III, Springer-Verlag, Berlin, 1982.

[13] S. Klavzˇar, J. Jerebic, and D.F. Rall, Distance-balanced graphs, Anna. Comb., 12, 71-79, (2008).

[14] S. Klavzˇar, A. Ajapakes, and I. Gutman, The Szeged and Wiener index of graphs, Appl. Math. Lett., 9 (5), 45-49,

(1996).

[15] M. Mirzargar, and A.R. Ashrafi, Some distance-based topologigcal indices of a non-commuting graph, Hacettepe J.Math. Stat. 41(4), 515-526, (2012).

[2] A. Azad, M. Eliasi, Distance in the non-commuting graph of groups, ARS Combin. 99, 279-287, (2011).

[3] A. Azad, Mohammad A. Iranmanesh, Cheryl E. Praeger and P. Spiga, Abelian coverings of finite general linear

groups and an application to their non-commuting graphs, J. Algebr. Comb. 34, 638-710, (2011).

[4] A. Azad, Cheryl E. Praeger, Maximal set of pairwise non-commuting elements of three-dimensional general linear

groups, Bull. Aust. Math. Soc. 0, 1-14, (2009).

[5] F. Buckley, and F. Harary, F. Distance in graphs, Addison-Wesley, Redwood, CA, 1990.

[6] D.M. Cvetkocic, M. Doob, H. Sachs, Spectar of graph theory and application, Academic Press, New York, 1980.

[7] P. Dankelmann, Average distance and independence numbers, Discrete Appl. Math., 51, 75-83, (1994).

[8] A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math.,

66, 211-249, (2001).

[9] A.A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publications De

L‘Institut math´ematiques Nouvelle s´erie, 56, 18-22, (1994).

[10] I. Gutman, A formula for the Wiener number of trees and its extension to graphs cycles, Graph theory Notes of

New York, 27, 9-15, (1994).

[11] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

[12] B. Huppert, and N. Blackburn, Finite groups III, Springer-Verlag, Berlin, 1982.

[13] S. Klavzˇar, J. Jerebic, and D.F. Rall, Distance-balanced graphs, Anna. Comb., 12, 71-79, (2008).

[14] S. Klavzˇar, A. Ajapakes, and I. Gutman, The Szeged and Wiener index of graphs, Appl. Math. Lett., 9 (5), 45-49,

(1996).

[15] M. Mirzargar, and A.R. Ashrafi, Some distance-based topologigcal indices of a non-commuting graph, Hacettepe J.Math. Stat. 41(4), 515-526, (2012).

[16] A.R. Moghaddamfar, W.J. Shi, W. Zhou and A.R. Zokayi, On the noncommuting graph associated with a finite

group, Siberian Math. J. 46 (2), 325-332, (2005).

group, Siberian Math. J. 46 (2), 325-332, (2005).

November 2014

Pages 105-115