Let G be a finite group and let cd(G) be the set of all complex irreducible
character degrees of G Let \rho(G) be the set of all primes which divide some
character degree of G. The prime graph \Delta(G) attached to G is a graph whose
vertex set is \rho(G) and there is an edge between two distinct primes u and v
if and only if the product uv divides some character degree of G. In this
paper, we show that if G is a finite group whose prime graph \Delta(G) has no
triangles, then \Delta(G) has at most 5 vertices. We also obtain a
classification of all finite graphs with 5 vertices and having no triangles
which can occur as prime graphs of some finite groups. Finally, we show that
the prime graph of a finite group can never be a cycle nor a tree with at least
5 vertices.Comment: 13 page
