{
  "abstract": "For an abelian group $Γ$ of order $n$, a graph $G$ of order $n$ is $Γ$-distance magic if it admits a bijection $V(G)\\toΓ$ whose open-neighborhood sums are constant, and group distance magic if this holds for every such $Γ$. More generally, for any finite abelian $Γ$, a $Γ$-magic map is a map $f\\colon V(G)\\toΓ$ with constant open-neighborhood sums; we call $f$ generating if its labels generate $Γ$, and affinely generating if its pairwise differences do. Let $G$ be regular, let $Γ\\cong\\mathbb{Z}/d_1\\oplus\\cdots\\oplus\\mathbb{Z}/d_r$ with $d_1\\mid\\cdots\\mid d_r$, put $B_Γ=\\bigoplus_{i<r}\\mathbb{Z}/d_i$, and let $\\overline{A}_m$ be the adjacency operator induced on $(\\mathbb{Z}/m)^{V(G)}$ modulo constants. We prove that $G$ admits a generating $Γ$-magic map iff $B_Γ\\hookrightarrow\\ker\\overline{A}_{d_r}$, and an affinely generating one iff $Γ\\hookrightarrow\\ker\\overline{A}_{d_r}$; when $|V(G)|=|Γ|$, it is $Γ$-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over $\\mathbb{Q}$, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating $(\\mathbb{Z}/3)^3$-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever $Γ$ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian $p$-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05934",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmet Batal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/501yn1yeb0rcm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05934",
  "fulltext_url": null,
  "id": "501yn1yeb0rcm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05934",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7212000142",
  "pdf_url": "https://arxiv.org/pdf/2609.05934",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/501yn1yeb0rcm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05934",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05934"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Algebraic characterizations of generating and affinely generating $Γ$-magic maps and $Γ$-distance magic labelings on regular graphs",
  "title_text": "Algebraic characterizations of generating and affinely generating Γ-magic maps and Γ-distance magic labelings on regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/501yn1yeb0rcm",
  "venue": "arXiv",
  "versions": []
}