Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Abstract

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 $ {\mathbb{A}_{10}} $ , L 4(8), L 4(4), and U 4(4). In this paper, we prove that if $ {\mathbb{A}_{16}} $ denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism $ {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} $ implies that $ {\mathbb{A}_{16}} \cong G $ .