Non-commuting graph of a group

Abstract

Let G be a non-abelian group and let Z ( G ) be the center of G. Associate a graph Γ G (called non-commuting graph of G) with G as follows: Take G \\ Z ( G ) as the vertices of Γ G and join two distinct vertices x and y, whenever x y ≠ y x . We want to explore how the graph theoretical properties of Γ G can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H , then | G | = | H | . Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that Γ G ≅ Γ H and | G | = | H | , then H is nilpotent.