On the noncommuting graph associated with a finite group

Let $G$ be a finite group. We define the noncommuting graph $\nabla(G)$ as follows: the vertex set of $\nabla(G)$ is $G\setminus Z(G)$ with two vertices $x$ and $y$ joined by an edge whenever the commutator of $x$ and $y$ is not the identity. We study some properties of $\nabla(G)$ and prove that, for many groups $G$, if $H$ is a group with $\nabla(G)$ isomorphic to $\nabla(H)$ then $|G|=|H|$.