@article { author = {Rezaei, R. and Varmazyar, M.}, title = {The graph of equivalence classes and Isoclinism of groups}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {43}, number = {6}, pages = {1801-1810}, year = {2017}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, abstract = {‎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 $V(\Gamma(G))=G\setminus Z(G)$ by taking $x\sim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)=\{u\in 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 $\{[x] \ | \ x \in G\setminus Z(G)\}$ 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.}, keywords = {Non-commuting graph,graph of equivalence classes,Isoclinism}, url = {http://bims.iranjournals.ir/article_1061.html}, eprint = {http://bims.iranjournals.ir/article_1061_2b7cba5aa6cdb3872025474bf8edf9fb.pdf} }