Graph product and the stability of circulant graphs

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.