Conditions on the edges and vertices of non-commuting graph

Abstract

Let G be a non- abelian finite group. The non-commuting graph of GG is defined as a graph with a vertex set G-Z(G)in which two vertices x and y are joined if and only if xy ? yx. We define GG=(V(GG), E(GG)) such that V(GG) is the vertices set and E(GG) is the edges set. In this paper, we invest some results on |E(GG)|, the degree of avertex of non-commuting graph and the number of conjugacy classes of a finite group. We found that that if GG ? GH is afinite group, then |G| = |H|.