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 ۳.۲۴ of [A . Abdollahi , S . Akbari , H . R , Maimani , Non-commuting graph of a group , J . Algebra , ۲۹۸ (۲۰۰۶) ۴۶۸-۴۹۲.] . We also prove that if G_۱ , G_۲ , ldots , G_n are finite groups such that Z(G_i)=۱ for i=۱ , ۲ , ldots , n and they are characterizable by non commuting graph , then G_۱ times G_۲ times cdots times G_n is characterizable by non-commuting graph .