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 article, 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.