Groups with the same non-commuting graph

The non-commuting graph Γ_G of a non-abelian group G is defined as follows. The vertex set of Γ_G is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that Γ_G≅Γ_H, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism Γ_G≅Γ_H implies G≅H.