Some results on characterization of finite group by non commuting graph

Abstract

The non commuting graph $nabla(G)$ of‎ ‎a non-abelian finite group $G$ is defined as follows‎: ‎its vertex set is‎ ‎$G‎- ‎Z (G)$ and two distinct vertices $x$ and $y$‎ ‎are joined by an edge if and only if the commutator of $x$ and $y$ is not the‎ ‎identity‎. ‎In this paper we prove some new results about this graph‎. ‎In‎ ‎particular we will give a new proof of Theorem 3.24 of [A‎. ‎Abdollahi‎, ‎S‎. ‎Akbari‎, ‎H‎. ‎R‎, ‎Maimani‎, ‎Non-commuting graph of a group‎, ‎<em>J‎. ‎Algebra</em>‎, ‎<strong>298</strong> (2006) 468-492.]‎. ‎We also prove that‎ ‎if $G_1‎, ‎G_2‎, ‎ldots‎, ‎G_n$ are finite groups such that $Z(G_i)=1$ for $i=1‎, ‎2‎,‎ldots‎, ‎n$‎ ‎and they are characterizable by non commuting graph‎, ‎then $G_1 times G_2‎ ‎times cdots times G_n$ is characterizable by non-commuting graph‎.