{
  "abstract": "A subset $C$ of the vertex set of a graph $Γ$ is said to be an $(a,b)$-regular set if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. A $(0,1)$-regular set is called a perfect code. Let $G$ be a finite group and let $α\\in\\operatorname{Aut}(G)$ be such that $α^2=\\mathrm{id}$. Let $ω_α(G)=\\{α(g^{-1})g:g\\in G\\}$. For a subset $S\\subseteq G$ satisfying $α(S)=S^{-1}$ and $S\\capω_α(G)=\\emptyset$, the generalized Cayley graph of $(G,α)$ with respect to $S$ is the graph with vertex set $G$ in which two distinct elements $g,h\\in G$ are adjacent if and only if $α(g^{-1})h\\in S$. In this paper, we prove two main results. First, let $H$ be an $α$-invariant subgroup of a finite group $G$, and let $P$ be an $α$-invariant Sylow 2-subgroup of $H$. We show that $H$ is a perfect code of $(G,α)$ if and only if $P[H,α]$ is a perfect code of $(G,α)$, where $[H,α]=\\langle h^{-1}α(h):h\\in H\\rangle$. Second, we consider finite groups $G$ possessing a cyclic subgroup $A$ of index 2 that is invariant under an involutory automorphism $α$. We show that $ω_α(G)$ is a subgroup of $A$. Using this result, we give a complete characterization of all pairs $(a,b)$ for which an $α$-invariant subgroup $H$ of $G$ is an $(a,b)$-regular set of $(G,α)$. In particular, our characterization applies to all finite groups possessing a characteristic cyclic subgroup of index 2, for every involutory automorphism $α$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00893",
  "author_total": 2,
  "authors": [
    {
      "name": "Yasamin Khaefi"
    },
    {
      "name": "Fateme Sadat Seiedali"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dv3yg2ed1wbs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00893",
  "fulltext_url": null,
  "id": "2dv3yg2ed1wbs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00893",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dv3yg2ed1wbs.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On subgroup perfect codes and regular sets in generalized Cayley graphs",
  "title_text": "On subgroup perfect codes and regular sets in generalized Cayley graphs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/2dv3yg2ed1wbs",
  "venue": "arXiv",
  "versions": []
}