Commuting graph of a group on a transversal

Abstract

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by $\mathcal{C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\ne y$ and $xy=yx$. Let $T$ be a transversal of the center $Z(G)$ of $G$. When $G$ is a finite non-abelian group and $X=T\setminus Z(G)$, we denote the graph $\mathcal{C}(G,X)$ by $\mathcal{T}(G)$. In this paper, we show that $\mathcal{T}(G)$ is a connected strongly regular graph if and only if $G$ is isoclinic to an extraspecial $2$-group of order at least $32$. We also characterize the finite non-abelian groups $G$ for which the graph $\mathcal{T}(G)$ is disconnected strongly regular.

14 pages, 4 figures