标题： 图积与循环图的稳定性 显示英文标题

标题： Graph product and the stability of circulant graphs

摘要： 若图 $\Gamma$ 满足 $\mathrm{Aut}(\Gamma\times K_2)\cong\mathrm{Aut}(\Gamma)\times \mathbb{Z}_{2}$，则称其为稳定的，否则称为不稳定的。 如果一个不稳定的图是连通的、非二部图，并且它的任意两个不同顶点具有不同的邻域，则称其为非平凡不稳定。 我们给出了保证各种图积（包括直积、直积束、Cartesian积、强积、半强积和字典序积）不稳定的条件。 受直积束不稳定性条件的启发，我们提出了一个关于循环图不稳定的新的充分条件。 这个条件生成了无穷多个不满足任何先前已知循环图不稳定性条件的非平凡不稳定的循环图。 显示英文摘要

摘要： A graph $\Gamma$ is said to be stable if $\mathrm{Aut}(\Gamma\times K_2)\cong\mathrm{Aut}(\Gamma)\times \mathbb{Z}_{2}$ and unstable otherwise. If an unstable graph is connected, non-bipartite and any two of its distinct vertices have different neighborhoods, then it is called nontrivially unstable. We establish conditions guaranteeing the instability of various graph products, including direct products, direct product bundles, Cartesian products, strong products, semi-strong products, and lexicographic products. Inspired by a condition for the instability of direct product bundles, we propose a new sufficient condition for circulant graphs to be unstable. This condition yields infinitely many nontrivially unstable circulant graphs that do not satisfy any previously established instability conditions for circulant graphs.