Characterization of A_16 by a noncommuting graph - Darafsheh, M. R.; Davoudi, Monfared M. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
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.