Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43620171130The graph of equivalence classes and Isoclinism of groups180118101061ENR.RezaeiDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer,
Iran.M.VarmazyarDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer,
Iran.Journal Article20150917Let $G$ 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.http://bims.iranjournals.ir/article_1061_2b7cba5aa6cdb3872025474bf8edf9fb.pdf