Characterization of the alternating group by its non-commuting graph

Abstract

Let G be any non-abelian group and Z(G) be its center. The non-commuting graph Γ G of G is the simple graph whose vertex set is G\Z(G), with two vertices x and y adjacent whenever xy≠xy. We prove that if Γ G is isomorphic to the non-commuting graph of the alternating group A n (n≥4), then G≅A n. This result together with a recent one due to Solomon and Woldar gives a complete positive answer to a conjecture proposed in [A. Abdollahi, S. Akbari, H.R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006) 468-492]: If S is any finite non-abelian simple group such that Γ S≅Γ G for some group G, then G≅S. © 2012 Elsevier Inc.