Automorphism groups of a class of cubic Cayley graphs on symmetric groups

Abstract

Let $S_n$ denote the symmetric group of degree $n$ with $n\ge 3$. Set $S=\{c_n=(12\dots n),c_n^{-1},(12)\}$. Let ${\Gamma }_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that ${\Gamma }_n$ ($n\ge 13$) is a normal Cayley graph, and that the full automorphism group of ${\Gamma }_n$ is equal to $\mathrm{Aut}({\Gamma }_n)=R(S_n)⋊⟨\mathrm{Inn}(\varphi )⟩\cong S_n⋊{\mathbb{Z}}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $\varphi =(12)(3n)(4n-1)(5n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(\varphi )$ is the inner isomorphism of $S_n$ induced by $\varphi$.

10 pages, 1 figure