On the Diameter of a p-Regular Conjugacy Class Graph of Finite Groups II

Abstract

Let Gbe a finite p-solvable group. Let us consider the graph Γ* p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.