Commuting graph of a group on a transversal

TL;DR

This paper characterizes when the commuting graph on a transversal of the center of a finite group is a connected or disconnected strongly regular graph, linking it to extraspecial 2-groups and group structure.

Contribution

It provides a complete characterization of the structure of the commuting graph on a transversal, identifying conditions for strong regularity and connectivity.

Findings

${ m extstyle extbf{T}}(G)$ is connected strongly regular iff $G$ is isoclinic to an extraspecial 2-group of order ≥ 32.

${ m extstyle extbf{T}}(G)$ is disconnected strongly regular for specific non-abelian groups.

The paper links graph properties to algebraic group structures, especially extraspecial 2-groups.

Abstract

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ 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 ${\cal C}(G,X)$ by ${\cal T}(G)$. In this paper, we show that ${\cal T}(G)$ 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 ${\cal T}(G)$ is disconnected strongly regular.