Stability of circulant graphs

TL;DR

This paper investigates the stability of circulant graphs, proving all odd prime order circulants are stable and providing counterexamples to a longstanding conjecture, thus advancing understanding of graph automorphisms.

Contribution

It proves all circulant graphs of odd prime order are stable and constructs infinitely many stable circulant graphs with compatible adjacency matrices, answering open questions.

Findings

All odd prime order circulant graphs are stable.

No arc-transitive nontrivially unstable circulant graphs exist.

Counterexamples to a 1989 conjecture are provided.

Abstract

The canonical double cover $\mathrm{D}(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\mathrm{Aut}(\mathrm{D}(\Gamma))=\mathrm{Aut}(\Gamma)\times\mathbb{Z}_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arc-transitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Maru\v{s}i\v{c}, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices.