On The automorphism groups of $us$-Cayley graphs

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 $0\ne g\in G$ 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 $us$-$Cayleygraph$, 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 $us$-$Cayleygraph$, then $Aut(\Gamma )=L(G)⋊A$, 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 Möbius ladders and $k$-ary $n$-cubes.

12 pages, 1 figure