Conjugacy classes of finite groups and graph regularity

Abstract

Given a finite group $G$, denote by $\Gamma (G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma (G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if $\Gamma (G)$ is a $k$-regular graph with $k\ge 1$, then $\Gamma (G)$ is a complete graph with $k+1$ vertices.