The graph of equivalence classes and Isoclinism of groups

‎Let G be a non-abelian group and let Γ(G) be the non-commuting graph of G‎. ‎In this paper we define an equivalence relation ∼ on the set of V(Γ(G))=G∖Z(G) by taking x∼y if and only if N(x)=N(y)‎, ‎where N(x)={u∈G | x and u are adjacent in Γ(G)} is the open neighborhood of x in Γ(G)‎. ‎We introduce a new graph determined by equivalence classes of non-central elements of G‎, ‎denoted ΓE(G)‎, ‎as the graph whose vertices are {[x] | x∈G∖Z(G)} and join two distinct vertices [x] and [y]‎, ‎whenever [x,y]≠1‎. ‎We prove that group G is AC-group if and only if ΓE(G) is complete graph‎. ‎Among other results‎, ‎we show that the graphs of equivalence classes of non-commuting graph associated with two isoclinic groups are isomorphic.