Non-commuting graph of a group

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph ΓG (called non-commuting graph of G) with G as follows: Take G\Z(G) as the vertices of ΓG and join two distinct vertices x and y, whenever xy≠yx. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that ΓG ΓH, then G=H. Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that ΓG ΓH and G=H, then H is nilpotent.