On the character degree graph of finite groups

TL;DR

This paper studies the structure of finite groups by analyzing the complement of their character degree graph, specifically characterizing groups where this complement is not bipartite, extending previous solvable case results.

Contribution

It characterizes finite groups with non-bipartite complements of their character degree graphs, extending prior work beyond solvable groups.

Findings

Identifies conditions for the complement graph to be non-bipartite

Extends previous results to non-solvable groups

Provides a classification of such groups

Abstract

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of [1], where the solvable case was treated.