On the regularity of a graph related to conjugacy class sizes of a normal subgroup

Abstract

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$ containing the element $x$. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not co-prime. In this article we prove that, if $\Gamma_G(N)$ is a connected incomplete regular graph, then $N= P \times{A}$ where $P$ is a $p$-group, for some prime $p$ and $A\leq{Z(G)}$, and ${\bf Z}(N)\not = N\cap {\bf Z}(G)$.