The canonical double cover $B(X)$ of a graph X is the direct product of X and $K_2$. If $\mathrm{\text{Aut}}(B(X))\cong \mathrm{\text{Aut}}(X)\times {\mathbb{Z}}_2$ then X is called stable; otherwise X is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. A circulant is a Cayley graph on a cyclic group. Qin et al. (2019) [18] conjectured that there are no nontrivially unstable circulants of odd order. In this paper we prove this conjecture.

中文翻译：

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循环的规范双封面

规范的双封面 $乙(X)$图X的直接积是X和${钾}_2$. 如果$\mathrm{\text{自动}}(乙(X))\cong \mathrm{\text{自动}}(X)\times {\mathbb{Z}}_2$则称X为稳定的；否则称X不稳定。如果一个不稳定的图是连通的、非二部的和不同的顶点有不同的邻域，那么它就是非平凡不稳定的。循环是循环群上的凯莱图。秦等人。(2019) [18] 推测不存在奇数阶的非平凡不稳定循环。在本文中，我们证明了这个猜想。