Regular sets in Cayley sum graphs on generalized dicyclic groups

Meiqi Peng; Yuefeng Yang; Meiqi Peng; Yuefeng Yang

doi:10.3934/math.20251326

AIMS Mathematics

2025, Volume 10, Issue 12: 30186-30205. doi: 10.3934/math.20251326

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Research article

Regular sets in Cayley sum graphs on generalized dicyclic groups

Meiqi Peng ,
Yuefeng Yang ^,

School of Science, China University of Geosciences, Beijing, China

Received: 23 October 2025 Revised: 29 November 2025 Accepted: 09 December 2025 Published: 24 December 2025
MSC : 05C25, 05C69, 94B25

For a graph $ \Gamma = (V(\Gamma), E(\Gamma)) $, a subset $ C $ of $ V(\Gamma) $ is called an $ (\alpha, \beta) $-regular set in $ \Gamma $, if every vertex of $ C $ is adjacent to exactly $ \alpha $ vertices of $ C $ and every vertex of $ V(\Gamma)\setminus C $ is adjacent to exactly $ \beta $ vertices of $ C $. In particular, if $ C $ is an $ (\alpha, \beta) $-regular set in some Cayley sum graph of a finite group $ G $ with connection set $ S $, then $ C $ is called an $ (\alpha, \beta) $-regular set of $ G $. In this paper, we considered a generalized dicyclic group $ G $ and for each subgroup $ H $ of $ G $, by giving an appropriate connection set $ S $, we determined each possibility for $ (\alpha, \beta) $ such that $ H $ is an $ (\alpha, \beta) $-regular set of $ G $.
- regular set,
- Cayley graph,
- Cayley sum graph,
- generalized dicyclic group,
- dicyclic group
Citation: Meiqi Peng, Yuefeng Yang. Regular sets in Cayley sum graphs on generalized dicyclic groups[J]. AIMS Mathematics, 2025, 10(12): 30186-30205. doi: 10.3934/math.20251326

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Abstract

For a graph $ \Gamma = (V(\Gamma), E(\Gamma)) $, a subset $ C $ of $ V(\Gamma) $ is called an $ (\alpha, \beta) $-regular set in $ \Gamma $, if every vertex of $ C $ is adjacent to exactly $ \alpha $ vertices of $ C $ and every vertex of $ V(\Gamma)\setminus C $ is adjacent to exactly $ \beta $ vertices of $ C $. In particular, if $ C $ is an $ (\alpha, \beta) $-regular set in some Cayley sum graph of a finite group $ G $ with connection set $ S $, then $ C $ is called an $ (\alpha, \beta) $-regular set of $ G $. In this paper, we considered a generalized dicyclic group $ G $ and for each subgroup $ H $ of $ G $, by giving an appropriate connection set $ S $, we determined each possibility for $ (\alpha, \beta) $ such that $ H $ is an $ (\alpha, \beta) $-regular set of $ G $.

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