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 \( {\mathbb{A}_{16}} \cong G \).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 11, pp. 1443–1450, November, 2010.
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Darafsheh, M.R., Davoudi Monfared, M. Characterization of \( {\mathbb{A}_{16}} \) by a noncommuting graph. Ukr Math J 62, 1673–1679 (2011). https://doi.org/10.1007/s11253-011-0459-2
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DOI: https://doi.org/10.1007/s11253-011-0459-2