Characterization of $A_{16}$ 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 $A_{10}, L_4(8), L_4(4)$, and $U_4(4)$. In this paper, we prove that if $A_{16}$ denotes the alternating group of degree 16; then, for any finite group $G$; the graph isomorphism $Γ_{A_{16}} ≅ Γ_G$ implies that $A_{16} ≅ G$