On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in 3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in 1],2],6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In 2],3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.