On the number of connected components of divisibility graph for certain simple groups

Abstract

The divisibility graph $mathscr{D}(G)$ for a finite group $G$ is a graph with vertex set $cs(G)setminus{1}$‎ ‎where $cs(G)$ is the set of conjugacy class sizes of $G$‎. ‎Two vertices $a$ and $b$ are adjacent whenever $a$ divides‎ ‎$b$ or $b$ divides $a$‎. ‎In this paper we will find the number of connected components of $mathscr{D}(G)$ where $G$ is a‎ ‎simple Zassenhaus group or an sporadic simple group‎.