A completeness criterion for the common divisor graph on $p$-regular class sizes

TL;DR

This paper proves that for a finite group, the common divisor graph on p-regular class sizes is complete if it is regular with degree at least one, establishing a clear structural characterization.

Contribution

It establishes a completeness criterion for the common divisor graph on p-regular class sizes in finite groups, showing regularity implies the graph is complete.

Findings

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

The graph's vertices form a complete graph with k+1 vertices.

The result characterizes the structure of the divisor graph under regularity conditions.

Abstract

Let $G$ be a finite group. For some fixed prime $p$, let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $\Gamma_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices.