The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined by equivalence classes of non-central elements of $G$‎, ‎denoted $Gamma_E(G)$‎, ‎as the graph whose vertices are ${[x] | x in Gsetminus Z(G)}$ and join two distinct vertices $[x]$ and $[y]$‎, ‎whenever $[x,y]neq 1$‎. ‎We prove that group $G$ is AC-group if and only if $Gamma_E(G)$ 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.