  In this work we discuss whether the non-commuting graph of a finite group can
determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani
conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting
graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2).
We pose a new conjecture (Conjecture 3) that, together with the assumption
$|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which
all centralizers of non-central elements are abelian.
