Groups with the same non-commuting graph

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