  Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The
non-commuting graph $\Gamma_G$ associated to $G$ is the graph whose vertex set
is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only
if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups
with irregular isomorphic non-commuting graphs, then $|G|=|H|$.
