Prime graphs of finite groups with a small number of triangles

The prime graph Delta(G) of a finite group G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper, we determine the upper bounds for the numbers of vertices of the prime graphs of finite groups which possess a small number of triangles. In some cases, we study the structure of such finite groups and their prime graphs in detail.