A conjecture related to groups with isomorphic non-commuting graphs

Let [Formula: see text] be a non-abelian group and [Formula: see text] be its center. The non-commuting graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and its two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Abdollahi, Akbari and Maimani conjectured that if [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text], then [Formula: see text]. They have further asked the question that for which group property [Formula: see text], if [Formula: see text] and [Formula: see text] are two non-abelian groups such that [Formula: see text], and [Formula: see text] has the group property [Formula: see text], then [Formula: see text] has also [Formula: see text]? In this paper, we prove the conjecture to be true for generalized quaternion groups and some classes of isoclinic groups. We also provide an affirmative answer to the question by assigning a group property [Formula: see text] to [Formula: see text] for some capable groups [Formula: see text] and showing that [Formula: see text].