Stability of pair graphs

TL;DR

This paper investigates the stability of pairs of graphs under direct product, providing necessary conditions and characterizations for stability and nontrivial instability, especially for connected, regular, and vertex-transitive graphs.

Contribution

It introduces the concept of $ ext{ } ext{-automorphisms} and characterizes nontrivially unstable graph pairs in specific cases, extending the understanding of graph automorphisms.

Findings

Necessary conditions for graph pair stability.

Characterization of nontrivially unstable pairs with regular, connected, vertex-transitive graphs.

Introduction of $ ext{-automorphisms} as a new automorphism concept.

Abstract

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(\Gamma,\Sigma)$ is stable if $Aut(\Gamma\times\Sigma) \cong Aut(\Gamma)\times Aut(\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. An unstable graph pair $(\Gamma,\Sigma)$ is said to be a nontrivially unstable graph pair if $\Gamma$ and $\Sigma$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pair of graphs to be stable. We also give a characterization of a pair of graphs $(\Gamma, \Sigma)$ to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and $\Sigma$ is vertex-transitive. This characterization is given in terms of the $\Sigma$-automorphisms of $\Gamma$, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph.