A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

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 $\mathrm{\mid }Z(G)\mathrm{\mid }\ge \mathrm{\mid }Z(H)\mathrm{\mid }$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.