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Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs

  • Chenhui Lv
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Abstract

Let G=Cay(ℤn,S) be a $3$-valent circulant digraph with connection set S={0,s,t}, where s,t∈ℤn* and t≠± s. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup US(ℤn)≤ℤn* that normalizes End(G). In this paper, we first establish necessary and sufficient conditions for a unit circulant digraph without $2$-cycles to admit an endomorphism whose image induces a directed cycle. Using this characterization, we explicitly determine US(ℤn) for the previously unresolved $0$-unit $3$-valent family.

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Cite

BibTeXRIS
BibTeX
@misc{lv2026automorphism,
  title = {{Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs}},
  author = {Chenhui Lv},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12318},
  eprint = {2609.12318},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12318},
}
RIS
TY  - UNPB
TI  - Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
AU  - Chenhui Lv
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12318
UR  - https://arxiv.org/abs/2609.12318
AB  - Let G=Cay(ℤ_n,S) be a $3$-valent circulant digraph with connection set S={0,s,t}, where s,t∈ℤ_n^* and t≠± s. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup U_S(ℤ_n)≤ℤ_n^* that normalizes End(G). In this paper, we first establish necessary and sufficient conditions for a unit circulant digraph without $2$-cycles to admit an endomorphism whose image induces a directed cycle. Using this characterization, we explicitly determine U_S(ℤ_n) for the previously unresolved $0$-unit $3$-valent family.
ER  -