Let G be a non-abelian group and Z(G) be the center of G. Associate a
graph ΞGβ (called non-commuting graph of G) with G as follows: take
GβZ(G) as the vertices of ΞGβ and join two distinct vertices
x and y, whenever xyξ =yx. Here, we prove that "the commutativity
pattern of a finite non-abelian p-group determine its order among the class
of groups"; this means that if P is a finite non-abelian p-group such that
ΞPββ ΞHβ for some group H, then β£Pβ£=β£Hβ£.Comment: to appear in Communications in Algebr