‎‎4‎‎-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 ρ(g). two primes p,q∈ρ(g) are adjacent in dg if and only if pq|a for some a∈cd(g). we determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.