On the Szeged and Eccentric connectivity indices of non-commutative graph of finite groups

Abstract

Let $G$ be a non-abelian group. <br />The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.<br /><br />In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|$. <br />Moreover, we determine eccentric connectivity index of $Gamma_G$ for every non-abelian finite group $G$ in terms of the number of conjugacy classes $k(G)$ and the size of the group $G$.