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 $(\mathrm{\Gamma },\mathrm{\Sigma })$ is stable if $\mathrm{Aut}(\mathrm{\Gamma }\times \mathrm{\Sigma })\cong \mathrm{Aut}(\mathrm{\Gamma })\times \mathrm{Aut}(\mathrm{\Sigma })$ and unstable otherwise, where $\mathrm{\Gamma }\times \mathrm{\Sigma }$ is the direct product of Γ and Σ. An unstable graph pair $(\mathrm{\Gamma },\mathrm{\Sigma })$ 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 $(\mathrm{\Gamma },\mathrm{\Sigma })$ 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.

我们开始研究一般图对的稳定性。这个概念是图形稳定性概念的概括。我们说一对图$（\mathrm{\Gamma }，\mathrm{\Sigma }）$ 如果稳定 $\mathrm{ut}（\mathrm{\Gamma }\times \mathrm{\Sigma }）\cong \mathrm{ut}（\mathrm{\Gamma }）\times \mathrm{ut}（\mathrm{\Sigma }）$ 否则就不稳定 $\mathrm{\Gamma }\times \mathrm{\Sigma }$是Γ和Σ的直接乘积。不稳定的图对$（\mathrm{\Gamma }，\mathrm{\Sigma }）$如果将Γ和Σ连接成互质图，则将其称为非平凡图对，并且它们中至少有一个是非二分图的，并且每个都具有不同顶点具有不同邻域的特性。我们获得了使一对图稳定的必要条件。我们还给出了一对图的特征$（\mathrm{\Gamma }，\mathrm{\Sigma }）$在两个图均被连接且具有互质数的正则规则且Σ是顶点传递的情况下，则是非平凡的不稳定。这种描述是根据Γ的Σ自同构给出的，它是本文引入的一个新概念，是图的自同构和双重自同构的推广。