Some finite groups with divisibility graph containing no triangles

Let G be a finite group. The graph D(G) is a divisibility graph of G. Its vertex set is the non-central conjugacy class sizes of G and there is an edge between vertices a and b if and only if a|b or b|a. In this paper, we investigate the structure of the divisibility graph D(G) for a non-solvable group with \sigma^{\ast}(G)=۲, a finite simple group G that satisfies the one-prime power hypothesis, a group of type(A),(B) or (C) and certain metacyclic p-groups and a minimal non-metacyclic p-group where p is a prime number. We will show that the divisibility graph D(G) for all of them has no triangles.

Keywords:

conjugacy class , divisibility graph , metacyclic