On symmetric Cayley graphs of valency thirteen

A Cayley graph $\text{Ga}=\text{Cay}(G,S)$ is said to be normal if the right-regular representation of $G$ is normal in $\text{Aut}\text{Ga}$. In this paper, we investigate the normality problem of the connected 13-valent symmetric Cayley graphs $\text{Ga}$ of finite nonabelian simple groups $G$, where the vertex stabilizer ${\text{A}}_v$ is soluble for $\text{A}=\text{Aut}\text{Ga}$ and $v\in V\text{Ga}$. We prove that $\text{Ga}$ is either normal or $G={\text{A}}_{12}$, ${\text{A}}_{38}$, ${\text{A}}_{116}$, ${\text{A}}_{207}$, ${\text{A}}_{311}$, ${\text{A}}_{935}$ or ${\text{A}}_{1871}$. Further, 13-valent symmetric non-normal Cayley graphs of ${\text{A}}_{38}$, ${\text{A}}_{116}$ and ${\text{A}}_{207}$ are constructed. This provides some more examples of non-normal 13-valent symmetric Cayley graphs of finite nonabelian simple groups since such graph (of valency 13) was first constructed by Fang, Ma and Wang in (J. Comb. Theory A 118, 1039–1051, 2011).