On automorphisms of the double cover of a circulant graph

Abstract

A graph $X$ is said to be "unstable" if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if it is unstable, connected, and nonbipartite, and no two distinct vertices of X have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n<8$, or $n=2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

25 pages (plus 8 pages of notes to aid the referee). Ancillary files include lists of the nontrivially unstable circulant graphs of order at most 50, together with MAGMA, sagemath, and Maple code to find these graphs

Cited by in corpus (1)

• Stability of Cayley graphs and Schur rings