Cycles and Bipartite Graph on Conjugacy Class of Groups

Abstract

Let $G$ be a finite non abelian group and $B(G)$ be the bipartite divisor graph of a finite group related to the conjugacy classes of $G$. We prove that $B(G)$ is a cycle if and only if $B(G)$ is a cycle of length 6 and $G\simeq A\times {SL}_2(q)$, where $A$ is abelian, and $q\in \{4,8\}$. We also prove that if $G/Z(G)$ is simple, where $Z(G)$ is the center of $G$, then $B(G)$ has no cycle of length 4 if and only if $G\simeq A\times {SL}_2(q)$, where $q\in \{4,8\}$.