On automorphisms of the double cover of a circulant graph

TL;DR

This paper investigates the automorphism properties of the canonical double cover of circulant graphs, identifying new conditions for instability and classifying all nontrivially unstable circulant graphs of order 2p, where p is prime.

Contribution

It introduces three new conditions that guarantee a circulant graph is unstable and classifies all nontrivially unstable circulant graphs of order 2p, expanding understanding of graph automorphisms.

Findings

Identified three new conditions implying instability in circulant graphs.

Classified all nontrivially unstable circulant graphs of order 2p for prime p.

Established criteria for the non-existence of nontrivially unstable circulant graphs based on order.

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 $4$.