The automorphism group of the Andr\'asfai graph

TL;DR

This paper determines the automorphism group of the Andr ext{a}sfai graph, showing it is isomorphic to a dihedral group, thus revealing its symmetry structure.

Contribution

The paper explicitly computes the automorphism group of the Andr ext{a}sfai graph, establishing its isomorphism with a dihedral group, which was previously unknown.

Findings

Automorphism group is isomorphic to the dihedral group n.

Provides a complete characterization of symmetries of Andre1sfai graphs.

Enhances understanding of the automorphism groups of Cayley graphs.

Abstract

Let $k \geq 1$ 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 $\mathbb{D}_{2n}$.