On the nilpotent conjugacy class graph of groups

Abstract

The nilpotent conjugacy class graph (or NCC-graph) of a group $G$ is a graph whose vertices are the nontrivial conjugacy classes of $G$ such that two distinct vertices $x^G$ and $y^G$ are adjacent if $\gen{x',y'}$ is nilpotent for some $x'\in x^G$ and $y'\in y^G$. We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph $\G_n(G)$ of the NCC-graph with vertices set $\{g^G\mid g\in G\setminus\Nil(G)\}$, where $\Nil(G)=\{g\in G\mid\gen{x,g}\text{ is nilpotent for all }x\in G\}$, and classify all finite non-nilpotent group $G$ with empty and triangle-free NCC-graphs.