$K_4$-free character graphs with diameter three

TL;DR

This paper investigates the structure of finite groups whose character graphs are $K_4$-free with diameter three, revealing a specific classification related to the first Janko sporadic simple group.

Contribution

It characterizes the structure of groups with $K_4$-free character graphs of diameter three, showing a unique form involving the Janko group and abelian factors.

Findings

If $| ho(G)|=5$, then $G$ is isomorphic to $J_1 imes A$ with $A$ abelian.

The paper establishes a precise condition linking the size of $ ho(G)$ to the group's structure.

It provides a classification of groups based on the properties of their character graphs.

Abstract

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\rm{cd}(G)$ be the set of all character degrees of $G$ and denote by $\rho(G)$ the set of primes which divide some character degrees in $\rm{cd}(G)$. The character 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 $G$. Suppose the character graph $\Delta(G)$ is $K_4$-free with diameter $3$. In this paper, we show that $|\rho(G)|\neq 5$, if and only if $G\cong J_1 \times A$, where $J_1$ is the first Janko's sporadic simple group and $A$ is abelian.