Conjugacy classes of finite groups and graph regularity

TL;DR

This paper investigates the structure of a graph constructed from a finite group's conjugacy class sizes, proving that if the graph is regular with degree at least one, it must be a complete graph.

Contribution

It establishes a classification result for the conjugacy class size graph, showing regularity implies completeness for nontrivial cases.

Findings

If the graph is k-regular with k ≥ 1, then it is a complete graph.

The graph's vertices correspond to noncentral conjugacy class sizes.

Regularity condition leads to a complete graph structure.

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 $k\geq 1$, then $\Gamma(G)$ is a complete graph with $k+1$ vertices.