Some results on characterization of finite group by non commuting graph

The non commuting graph 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 [2]. We also prove that if $G_1$، $G_2$،…، $G_n$ are finite groups such that $Z (G_i) =1$ for $i=1،2،…،n$ and they are characterizable by non commuting graph، then $G_1times…times G_n$ is characterizable by non commuting graph.