Stability of graph pairs

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 ( Γ , Σ ) is stable if Aut ( Γ × Σ ) ≅ Aut ( Γ ) × Aut ( Σ ) and unstable otherwise, where Γ × Σ is the direct product of Γ and Σ. An unstable graph pair ( Γ , Σ ) is said to be a nontrivially unstable graph pair if Γ and Σ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighborhoods. We obtain necessary conditions for a pair of graphs to be stable. We also give a characterization of a pair of graphs ( Γ , Σ ) to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and Σ is vertex-transitive. This characterization is given in terms of the Σ-automorphisms of Γ, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph.