On the nilpotent conjugacy class graph of groups

The nilpotent conjugacy class graph (or NCC-graph) of a group is a graph whose vertices are the nontrivial conjugacy classes of such that two distinct vertices and are adjacent if is nilpotent for some and . We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph of the NCC-graph with vertices set , where , and classify all finite non-nilpotent group with empty and triangle-free NCC-graphs