Year: 2025
Type: article
Abstract: 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 $e \ge 1$.
DOI: 10.37236/13327
Authors: Ademir Hujdurović, István Kovács
Institution: University of Primorska
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Awards: The Weiss Conjecture and Generalisations, Distance-regular graphs: irreducible T-modules with endpoint 1 and action of the automorphism group, Mathematical modelling and encryption: from theoretical concepts to real-life applications, Symmetries in graphs via rigid cells, Terwilliger algebra of a graph, Geometries, Graphs, Groups and their Links, Designing certain discrete mathematical objects in spectral domain, Automorphisms and Isomorphisms of Finite Graphs, Topics in structural graph teory: invariants, separators and intersection representations, Discrete combinatorial objects in the spectral domain - intersection analysis, Algebra, discrete mathematics, probability and game theory
OpenAlex ID: W4411450231