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, J. Algebra, 298 (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.
A. Abdollahi, S. Akbari, H. R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298, 468-492 M. R. Darafsheh (2009). Groups with the same non-commuting graph. Discrete Appl.
Math.. 157 (4), 833-837 Ron Solomon and Andrew Woldar (2012). All Simple groups are characterized by their
non-commuting graphs. preprint.