SOME RESULT ON NILPOTENT CONJUGACY CLASS GRAPH OF GROUPS

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 hx 0 , y 0 i is nilpotent for some x 0 ∈ x G and y 0 ∈ 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 Γ n (G) of the NCC-graph with vertices set {g G |g ∈ G \ Nil(G)}, where Nil(G) = {g ∈ G : hx, gi is nilpotent for all x ∈ G}, and classify all finite non-nilpotent group G with empty and triangle-free NCC-graphs.