Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On automorphisms of the double cover of a circulant graph   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 $4$.

30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th