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

Abstract

‎Let $G$ be a finite group‎. ‎The prime degree graph of $G$‎, ‎denoted‎ ‎by $Delta(G)$‎, ‎is an undirected graph whose vertex set is $rho(G)$ 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 $G$‎, ‎$Delta(G)$ is a connected graph which has only one triangle and four vertices‎.