Stability of Cayley graphs on abelian groups of odd order

TL;DR

This paper proves that certain Cayley graphs on odd-order abelian groups are stable, meaning their automorphisms are only those induced by automorphisms of the graph and its factors, using a short and elementary proof.

Contribution

It establishes the stability of Cayley graphs on odd-order abelian groups and generalizes the result to products with a broader class of graphs.

Findings

Cayley graphs on odd-order abelian groups are stable.

The stability is shown via the automorphism properties of the direct product with K_2.

The proof is short, elementary, and extends to more general connected graphs.

Abstract

Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X \times K_2$ (also called the "canonical double cover" of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is "stable". The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.