# 4-Regular prime graphs of nonsolvable groups

## TL;DR

This paper characterizes which simple 4-regular graphs can serve as prime graphs of finite nonsolvable groups based on their character degree sets.

## Contribution

It identifies the specific simple 4-regular graphs that can be realized as prime graphs of finite nonsolvable groups.

## Key findings

- Characterization of 4-regular prime graphs for nonsolvable groups
- Identification of possible simple 4-regular graphs as prime graphs
- Contribution to understanding the structure of character degree sets in group theory

## Abstract

Let $G$ be a finite group and $\text{cd}(G)$ denote the character degree set for $G$. The prime graph $\Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $\text{cd}(G)$, denoted $\rho(G)$. Two primes $p,q\in \rho(G)$ are adjacent in $\Delta(G)$ if and only if $pq|a$ for some $a\in \text{cd}(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.