{"title":"On the automorphism groups of us-Cayley graphs","authors":"S. Mirafzal","doi":"10.26493/2590-9770.1624.a3d","DOIUrl":null,"url":null,"abstract":"Let $G$ be a finite abelian group written additively with identity $0$, and $\\Omega$ be an inverse closed generating subset of $G$ such that $0\\notin \\Omega$. We say that $ \\Omega $ has the property \\lq\\lq{}$us$\\rq\\rq{} (unique summation), whenever for every $0 \\neq g\\in G$ if there are $s_1,s_2,s_3, s_4 \\in \\Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have $\\{s_1,s_2 \\} = \\{s_3,s_4 \\}$. We say that a Cayley graph $\\Gamma=Cay(G;\\Omega)$ is a $us$-$Cayley\\ graph$, whenever $G$ is an abelian group and the generating subset $\\Omega$ has the property \\lq\\lq{}$us$\\rq\\rq{}. In this paper, we show that if $\\Gamma=Cay(G;\\Omega)$ is a $us$-$Cayley\\ graph$, then $Aut(\\Gamma)=L(G)\\rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $\\theta$ of the group $G$ such that $\\theta(\\Omega)=\\Omega$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including M\\\"{o}bius ladders and $k$-ary $n$-cubes.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"7 1","pages":"1"},"PeriodicalIF":0.0000,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Art of Discrete and Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/2590-9770.1624.a3d","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 5
Abstract
Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation), whenever for every $0 \neq g\in G$ if there are $s_1,s_2,s_3, s_4 \in \Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have $\{s_1,s_2 \} = \{s_3,s_4 \}$. We say that a Cayley graph $\Gamma=Cay(G;\Omega)$ is a $us$-$Cayley\ graph$, whenever $G$ is an abelian group and the generating subset $\Omega$ has the property \lq\lq{}$us$\rq\rq{}. In this paper, we show that if $\Gamma=Cay(G;\Omega)$ is a $us$-$Cayley\ graph$, then $Aut(\Gamma)=L(G)\rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $\theta$ of the group $G$ such that $\theta(\Omega)=\Omega$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including M\"{o}bius ladders and $k$-ary $n$-cubes.