On The automorphism groups of $us$-Cayley graphs

TL;DR

This paper characterizes the automorphism groups of a class of Cayley graphs called $us$-Cayley graphs, showing they are semidirect products of the group with automorphisms preserving the generating set, and applies this to specific graph classes.

Contribution

It provides a complete description of automorphism groups for $us$-Cayley graphs, a new class defined by a unique summation property, and applies this to known graph families.

Findings

Automorphism group of $us$-Cayley graphs is $L(G) times A$.

Explicit automorphism groups for Möbius ladders.

Explicit automorphism groups for $k$-ary $n$-cubes.

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.