A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $\mathrm{Aut}(\Gamma \times K_2)$ is not isomorphic to $\mathrm{Aut}(\Gamma) \times \mathbb{Z}_2$. We show that a connected and non-bipartite Cayley graph $\mathrm{Cay}(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ satisfying certain properties. The S-rings with these properties are characterized if $H$ is a cyclic group of twice odd order. As an application, a necessary and sufficient condition is given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$.
@article{10_37236_13327,
author = {Ademir Hujdurovi\'c and Istv\'an Kov\'acs},
title = {Stability of {Cayley} graphs and {Schur} rings},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {2},
doi = {10.37236/13327},
zbl = {1569.05137},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13327/}
}
TY - JOUR
AU - Ademir Hujdurović
AU - István Kovács
TI - Stability of Cayley graphs and Schur rings
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/13327/
DO - 10.37236/13327
ID - 10_37236_13327
ER -
%0 Journal Article
%A Ademir Hujdurović
%A István Kovács
%T Stability of Cayley graphs and Schur rings
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/13327/
%R 10.37236/13327
%F 10_37236_13327
Ademir Hujdurović; István Kovács. Stability of Cayley graphs and Schur rings. The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/13327
