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 ${\mathrm{\Gamma }}_G$ as follows: Take $G\setminus Z(G)$ as vertices of ${\mathrm{\Gamma }}_G$ and join two distinct vertices $x$ and $y$ whenever $xy\ne yx$. Graph ${\mathrm{\Gamma }}_G$ is called the non-commuting graph of $G$ and many of graph theoretical properties of ${\mathrm{\Gamma }}_G$ have been studied. In this paper, we study some metric graph properties of ${\mathrm{\Gamma }}_G$.