ON THE SZEGED INDEX OF NON-COMMUTATIVE GRAPH OF GENERAL LINEAR GROUP

Abstract

Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is a<br />graph $Gamma_G$ as follows: Take $Gsetminus Z(G)$ as vertices of<br />$Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever<br />$yxneq yx$. $Gamma_G$ is called the non-commuting graph of $G$.<br /> In recent years many interesting works have been done in non-commutative graph of groups.<br /> Computing the clique number, chromatic number, Szeged index and Wiener index play important role in graph theory. In particular, the clique<br /> number of non-commuting graph of some the general linear groups has been determined.<br /><br /> nt Recently, Wiener and Szeged indices<br />have been computed for $Gamma_{PSL(2,q)}$, where $qequiv 0 (mod<br />~~4)$. In this paper we will compute the Szeged index for<br />$Gamma_{PSL(2,q)}$, where $qnotequiv 0 (mod ~~ 4)$.