Stability of graph pairs

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