Distance in the non-commuting graph of groups

Abstract

Let G be a non-abelian group and let Z(G) be the center of G. Associate with G a graph ΓG as follows: Take G\Z(G) as vertices of ΓG and join two distinct vertices x and y whenever xy ≠ yx. Graph ΓG is called the non-commuting graph of G and many of graph theoretical properties of ΓG have been studied. In this paper we study some metric graph properties of ΓG.