Spectral Properties of $NC$-Graphs

M. Ghorbani, Z. Gharavi-Alkhansari

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph ($NC$-graph) $\Gamma(G)$ of the group $G$ is a graph with the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent whenever $xy \neq yx$. The aim of this paper is to prove that for given group $G$, $\frac{G}{Z(G)}\cong \Bbb{Z}_p×\Bbb{Z}_p$ if and only if $\Gamma(G)$ is a regular ($p+1$)-partite graph. Also we consider the isomorphism of the non-commuting graph with some special graphs.