Stability of graph pairs involving vertex-transitive graphs

Abstract

A pair of graphs $(\Gamma ,\Sigma )$ is said to be stable if the full automorphism group of $\Gamma \times \Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where $\Gamma \times \Sigma$ is the direct product of $\Gamma$ and $\Sigma$. In this paper, we reduce the study of the stability of any pair of regular graphs $(\Gamma ,\Sigma )$ with coprime valencies and vertex-transitive $\Sigma$ to that of $(\Gamma ,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(\Gamma ,\Sigma )$ with coprime valencies in the case when $\Sigma$ is vertex-transitve and the stability of $(\Gamma ,K_2)$ is known.

9 pages