‎$‎۴‎$‎-Regular prime graphs of nonsolvable groups

Let $G$ be a finite group and $\cd(G)$ denote the character degree set for $G$. The prime graph $\DG$ is a simple graph whose vertex set consists of prime divisors of elements in $\cd(G)$, denoted $\rho(G)$. Two primes $p,q\in \rho(G)$ are adjacent in $\DG$ if and only if $pq|a$ for some $a\in \cd(G)$. We determine which simple ۴-regular graphs occur as prime graphs for some finite nonsolvable group.

Keywords:

Nonsolvable group , character , character degree , graph , prime graph