31 خرداد 1403
رشيد رضايي

رشید رضایی

مرتبه علمی: دانشیار
نشانی:
تحصیلات: دکترای تخصصی / ریاضی محض
تلفن:
دانشکده: دانشکده علوم ریاضی و آمار

مشخصات پژوهش

عنوان
THE GRAPH OF EQUIVALENCE CLASSES AND ISOCLINISM OF GROUPS
نوع پژوهش مقاله چاپ شده
کلیدواژه‌ها
Non-commuting graph, graph of equivalence classes, Isoclinism.
سال
2017
مجله Bulletin Of The Iranian Mathematical Society
پژوهشگران رشید رضایی

چکیده

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.