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 ΓG (called non-commuting graph of G) with G as follows: Take G \ Z ( G ) as the vertices of ΓG and join two distinct vertices x and y, whenever x y ≠ y x. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that ΓG ≅ ΓH, 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 ΓG ≅ ΓH and | G | = | H |, then H is nilpotent. © 2006 Elsevier Inc. All rights reserved.