Stability of Cayley graphs and Schur rings

TL;DR

This paper characterizes the stability of connected non-bipartite Cayley graphs using Schur rings, providing new criteria and applications for understanding automorphism groups and graph twins.

Contribution

It establishes a link between Cayley graph stability and Schur rings, characterizes relevant Schur rings for certain groups, and applies these results to circulant graphs and known theorems.

Findings

Unstable Cayley graphs correspond to specific Schur rings over $H imes bZ_2$.

Characterization of Schur rings for abelian groups of odd order and cyclic groups of twice odd order.

Conditions for instability of circulant graphs of order $2p^e$.

Abstract

A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $Aut(\Gamma \times K_2)$ is not isomorphic to $Aut(\Gamma) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph $Cay(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ having certain properties. The Schur rings with these properties are characterized if $H$ is an abelian group of odd order or a cyclic group of twice odd order. As an application, a short proof is given for the result of Witte Morris stating that every connected unstable Cayley graph on an abelian group of odd order has twins (Electron.~J.~Combin, 2021). As another application, sufficient and necessary conditions are given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$.