Characterization of by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ G ; called the noncommuting graph of G ; with a vertex set G − Z ( G ) such that two vertices x and y are adjacent if and only if xy ≠ yx : Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ S ≅ Γ G ; then S ≅ G : It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except , L 4 (8), L 4 (4), and U 4 (4). In this paper, we prove that if denotes the alternating group of degree 16; then, for any finite group G ; the graph isomorphism implies that .