Character graphs with diameter three

TL;DR

This paper investigates the structure of character graphs of finite groups with diameter three, showing their complements are bipartite and characterizing their structure.

Contribution

It establishes a new structural property of character graphs with diameter three and determines their form, advancing understanding of group character graphs.

Findings

Complement of the graph is bipartite when diameter is three

Character graph structure is explicitly determined in this case

Provides new insights into the relationship between graph diameter and structure

Abstract

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that if the diameter of $\Delta(G)$ is equal to three, then the complement of $\Delta(G)$ is bipartite. Also in this case, we determine the structure of the character graph $\Delta(G)$.