Rezaei, R., Varmazyar, M. (2017). The graph of equivalence classes and Isoclinism of groups. Bulletin of the Iranian Mathematical Society, 43(6), 1801-1810.

R. Rezaei; M. Varmazyar. "The graph of equivalence classes and Isoclinism of groups". Bulletin of the Iranian Mathematical Society, 43, 6, 2017, 1801-1810.

Rezaei, R., Varmazyar, M. (2017). 'The graph of equivalence classes and Isoclinism of groups', Bulletin of the Iranian Mathematical Society, 43(6), pp. 1801-1810.

Rezaei, R., Varmazyar, M. The graph of equivalence classes and Isoclinism of groups. Bulletin of the Iranian Mathematical Society, 2017; 43(6): 1801-1810.

The graph of equivalence classes and Isoclinism of groups

^{}Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.

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.

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