论文标题
一对图的稳定性
Stability of pair graphs
论文作者
论文摘要
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，如果$ aut（γ\timesς）\ cong aut（γ）\ times aut（σ）$（σ）$且不稳定，则一对图$（γ，σ）$是稳定的，其中$γ\timesς$是$γ$和$σ$的直接产品。如果$γ$和$σ$是连接的coprime图，则不稳定的图对$（γ，σ）$被认为是非平稳的图对，至少其中一个是非双分部分，并且每个人都有不同顶点具有不同社区的属性。我们获得了一对图稳定的必要条件。我们还给出了一对图$（γ，σ）$的表征，在两个图形连接并使用副本率进行常规的情况下，在不稳定的情况下是不稳定的，而$σ$是顶点传递的。这种表征是根据$σ$的$γ$的$σ$ - 自动形态，这是本文中引入的一个新概念，作为对图的自动形态和两倍的自动形态的概括。
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 $Aut(Γ\timesΣ) \cong Aut(Γ)\times Aut(Σ)$ and unstable otherwise, where $Γ\timesΣ$ 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 neighbourhoods. 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.
