Groups with the same non-commuting graph. (Q1028482)

Let \(G\) be a non-Abelian group and associate a non-commuting graph \(\nabla(G)\) with \(G\) as follows: the vertex set of \(\nabla(G)\) is \(G\setminus Z(G)\) where \(Z(G)\) denotes the center of \(G\) and two vertices \(x\) and \(y\) are adjacent if and only if \(xy\neq yx\). The non-commuting graph \(\nabla(G)\) was first considered by Paul Erdős in 1975 [see \textit{B. H. Neumann}, J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. It has been conjectured that if \(G\) and \(H\) are two non-Abelian finite groups such that \(\nabla(G)\cong\nabla(H)\), then \(|G|=|H|\) [see \textit{A. Abdollahi, S. Akbari} and \textit{H. R. Maimani}, J. Algebra 298, No. 2, 468-492 (2006; Zbl 1105.20016)]. But this conjecture is not correct [see \textit{A. R. Moghaddamfar}, Sib. Mat. Zh. 47, No. 5, 1112-1116 (2006); translation in Sib. Math. J. 47, No. 5, 911-914 (2006; Zbl 1139.20019)]. Moreover, in the case that \(H\) is a simple group the above conjecture implies \(G\cong H\). In this paper, the author's aim is to prove this case of the conjecture for all the finite non-Abelian simple groups \(H\). Then for certain simple groups \(H\) whose Gruenberg-Kegel graphs are not connected, the author shows that the graph isomorphism \(\nabla(G)\cong\nabla(H)\) implies \(G\cong H\).