On nonsolvable groups whose prime degree graphs have four vertices and one triangle

‎Let G be a finite group‎. ‎The prime degree graph of G‎, ‎denoted‎ ‎by Δ(G)‎, ‎is an undirected graph whose vertex set is ρ(G) and there is an edge‎ ‎between two distinct primes p and q if and only if pq divides some irreducible‎ ‎character degree of G‎. ‎In general‎, ‎it seems that the prime graphs‎ ‎contain many edges and thus they should have many triangles‎, ‎so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles‎. ‎In this paper we consider the case where for a nonsolvable group G‎, ‎Δ(G) is a connected graph which has only one triangle and four vertices‎.