{
  "abstract": "The center of distances $C(Y)$ of a metric space $(Y,ρ)$ is the set of all $t\\geq 0$ for which the equation $ρ(y,p)=t $ has a solution for each point $p\\in Y$. We say that a simple graph $CG_Y$ is the central graph of a metric space $(Y,ρ)$ if $Y$ is the vertex set of $CG_Y$ and distinct vertices $x,y\\in Y$ are adjacent if and only if $ρ(x,y)\\in C(Y)$. We prove the inequality $|E(CG_Y)| \\geq \\left\\lceil \\frac{1}{2}|Y| \\right\\rceil $ for all finite metric spaces $(Y,ρ)$ with $|Y|\\geq 2$ and non-empty $CG_Y.$ It is also proved that the inequality $|E(CG_X)| \\geq |X|-1 $ holds for each finite ultrametric space $(X,d)$. The central graphs of metric spaces $(Y,ρ)$ and ultrametric spaces $(X,d)$ satisfying $|E(CG_Y)|=\\left\\lceil \\frac{1}{2}|Y| \\right\\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11596",
  "author_total": 2,
  "authors": [
    {
      "name": "Oleksiy Dovgoshey"
    },
    {
      "name": "Olga Rovenska"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eb5qcd19a7m9q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11596",
  "fulltext_url": null,
  "id": "eb5qcd19a7m9q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11596",
  "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.GN"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221156163",
  "pdf_url": "https://arxiv.org/pdf/2610.11596",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eb5qcd19a7m9q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11596",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11596"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Graph Corresponding to Center of Distances of Metric Space",
  "title_text": "On the Graph Corresponding to Center of Distances of Metric Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/eb5qcd19a7m9q",
  "venue": "arXiv",
  "versions": []
}