Finite groups whose prime graphs are regular

TL;DR

This paper characterizes finite groups whose prime graphs are 3-regular, proving they are complete graphs with four vertices, thus linking group structure to graph regularity.

Contribution

It establishes a precise characterization of finite groups with 3-regular prime graphs, showing they are exactly the groups with a complete prime graph of four vertices.

Findings

Prime graph of a finite group is 3-regular if and only if it is a complete graph with four vertices.

Provides a complete classification of groups with 3-regular prime graphs.

Connects the structure of finite groups to properties of their associated prime graphs.

Abstract

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime graph \Delta(G) associated to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. In this paper, we show that the prime graph \Delta(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices.