The graph of equivalence classes and Isoclinism of groups

Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Malayer University‎, ‎Malayer‎, ‎Iran.


‎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.


Main Subjects

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