Stability of graph pairs involving vertex-transitive graphs

A pair of graphs (Γ,Σ) is said to be stable if the full automorphism group of Γ×Σ is isomorphic to the product of the full automorphism groups of Γ and Σ and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. In this paper, we reduce the study of the stability of any pair of regular gra...

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Bibliographic Details
Published in:Discrete mathematics 2024-04, Vol.347 (4), p.113856, Article 113856
Main Authors: Qin, Yan-Li, Xia, Binzhou, Zhou, Sanming
Format: Article
Language:English
Subjects:
Direct product of graphs
Stable graph
Stable graph pair
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Summary:A pair of graphs (Γ,Σ) is said to be stable if the full automorphism group of Γ×Σ is isomorphic to the product of the full automorphism groups of Γ and Σ and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. In this paper, we reduce the study of the stability of any pair of regular graphs (Γ,Σ) with coprime valencies and vertex-transitive Σ to that of (Γ,K2). Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs (Γ,Σ) with coprime valencies in the case when Σ is vertex-transitive and the stability of (Γ,K2) is known.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2023.113856