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 equivalence graph of the comaximal graph of a group
نوع پژوهش مقاله چاپ شده
کلیدواژه‌ها
comaximal graph, equivalence graph, complete graph, planar graph, perfect graph.
سال
2020
مجله Quasigroups and Related Systems
پژوهشگران رشید رضایی

چکیده

Let $G$ be a finite group. The comaximal graph of $G$, denoted by $\Gamma_m(G)$, is a graph whose vertices are the proper subgroups of $G$ that are not contained in the Frattini subgroup of $G$ and join two distinct vertices $H$ and $K$, whenever $G=\langle H,K \rangle$. In this paper, we define an equivalence relation $\sim$ on $V(\Gamma_m(G))$ by taking $H \sim K$ if and only if their open neighborhoods are the same. We introduce a new graph determined by equivalence classes of $V(\Gamma_m(G))$, denoted $\Gamma_E(G)$, as follows. The vertices are $V(\Gamma_E(G))=\{[H]|H\in V(\Gamma_m(G))\}$ and two equivalence classes $[H]$ and $[K]$ are adjacent in $\Gamma_E(G)$ if and only if $H$ and $K$ are adjacent in $\Gamma_m(G)$. We will state some basic graph theoretic properties of $\Gamma_E(G)$ and study the relations between some properties of graph $\Gamma_m(G)$ and $\Gamma_E(G)$, such as the chromatic number, clique number, girth and diameter. Moreover, we classify the groups for which $\Gamma_E(G)$ is complete, regular or planar. Among other results, we show that if the number of maximal subgroups of the group $G$ is less or equal than 4, then $\Gamma_m(G)$ and $\Gamma_E(G)$ are perfect graphs.