Canonical double covers of circulants.

The canonical double cover B (X) of a graph X is the direct product of X and K 2. If Aut (B (X)) ≅ Aut (X) × 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. [ABSTRACT FROM AUTHOR]