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.