Finite groups with graphs containing no triangles

Let G be a non-Abelian finite group, Γ(G) the attached graph related to its conjugacy classes. The aim of this paper is to prove that the symmetric group S3, the dihedral group D5, the three pairwise non-isomorphic non-Abelian groups of order 12, and the non-Abelian group T21 of order 21, is the complete list of all G such that Γ(G) contains no triangles.