Stability of graph pairs involving vertex-transitive graphs

Abstract

A pair of graphs (Γ,Σ) is said to be stable if the full automorphism group of Γ×Σ is isomorphic to the product of the full automorphism groups of Γ and Σ and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. In this paper, we reduce the study of the stability of any pair of regular graphs (Γ,Σ) with coprime valencies and vertex-transitive Σ to that of (Γ,K2). Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs (Γ,Σ) with coprime valencies in the case when Σ is vertex-transitive and the stability of (Γ,K2) is known.