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