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 characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 for finite AC-groups whenever |Z(G)|≥|Z(H)|.