Commuting conjugacy class graphs of finite groups

Abstract

Suppose that G is a finite non-abelian group. Define the graph Γ(G) with the non-central conjugacy classes of G as vertex set and two distinct vertices A and B are adjacent if and only if there are x∈A and y∈B such that xy=yx. The graph Γ(G) is called the commuting conjugacy class graph of G and introduced by Mohammadian et al. in  [A. Mohammadian, A. Erfanian, M. Farrokhi D. G. and B. Wilkens,  Triangle-free commuting conjugacy class graphs, {J. Group Theory} {19} (3) (2016) 1049--1061]. In this paper, the graph structure of the commuting conjugacy class graph of nilpotent groups of order n are obtained in which n is not divisible by p5, for every prime factor p of n.