  Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a
graph $\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take
$G\setminus Z(G)$ as the vertices of $\Gamma_G$ and join two distinct vertices
$x$ and $y$, whenever $xy\neq yx$. Here, we prove that "the commutativity
pattern of a finite non-abelian $p$-group determine its order among the class
of groups"; this means that if $P$ is a finite non-abelian $p$-group such that
$\Gamma_P\cong \Gamma_H$ for some group $H$, then $|P|=|H|$.
