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