The automorphism group of the Andr\'asfai graph

Let $k\ge 1$ be an integer and $n=3k-1$. Let ${\mathbb{Z}}_n$ denote the additive group of integers modulo $n$ and let $C$ be the subset of ${\mathbb{Z}}_n$ consisting of the elements congruent to 1 modulo 3. The Cayley graph $Cay({\mathbb{Z}}_n;C)$ is known as the Andr$\acute{a}$sfai graph And($k$). In this note, we determine the automorphism group of this graph. We will show that $Aut(And(k))$ is isomorphic with the dihedral group ${\mathbb{D}}_{2n}$.