ON GRAPHS RELATED TO CONJUGACY CLASSES OF GROUPS

Let G be a finite group. Attach to G the following two graphs: Gamma - its vertices are the non-central conjugacy classes of G, and two vertices are connected if their sizes are not coprime, and Gamma* - its vertices are the prime divisors of sizes of conjugacy classes of G, and two vertices are connected if they both divide the size of some conjugacy class of G. We prove that whenever Gamma* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Gamma* is disconnected if and only if G 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 Gamma is also at most 3, whenever the graph is connected, and that Gamma is disconnected if and only if G is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Gamma and Gamma* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible.