Stability of graph pairs involving vertex-transitive graphs

TL;DR

This paper investigates the stability of pairs of vertex-transitive graphs, reducing the problem to a well-studied case involving the graph K2, thereby enabling easier determination of stability for broader classes.

Contribution

It reduces the stability analysis of regular vertex-transitive graph pairs with coprime valencies to the case involving K2, simplifying the problem significantly.

Findings

Stability of graph pairs can be reduced to the case involving K2.

Determines stability for pairs with coprime valencies when one graph is vertex-transitive.

Provides a method to analyze stability using known results for K2.

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.