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 σ∗(g) = 2, 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.