Finite groups whose prime graphs are regular

Let G   be a finite group and let Irr(G)Irr(G) be the set of all irreducible complex characters of G  . Let cd(G)cd(G) be the set of all character degrees of G   and denote by ρ(G)ρ(G) the set of primes which divide some character degrees of G  . The prime graph Δ(G)Δ(G) associated to G   is a graph whose vertex set is ρ(G)ρ(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  . In this paper, we show that the prime graph Δ(G)Δ(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices.