{
  "abstract": "In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\\mathbb Z_{pq}$, where $p$ and $q$ are distinct odd primes with $q>p$. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of $BS(Γ(\\mathbb Z_{pq}))$ that form metric bases when $q\\geq 2p-1$. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate $M$-subdivision graphs obtained by subdividing selected edges of $Γ(\\mathbb Z_{pq})$. In addition to the lower bounds for subdivisions of $p-3$ and $p-2$ edges, we prove an exact formula for matching subdivisions of arbitrary size $r$, $0\\leq r\\leq p-2$, namely $\\dim(G_r)=p+q-r-4$. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09896",
  "author_total": 4,
  "authors": [
    {
      "name": "Vidya S"
    },
    {
      "name": "Prasanna Poojary"
    },
    {
      "name": "Bilal Ahmad Rather"
    },
    {
      "name": "Vadiraja Bhatta G R"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f8g38f07a185c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09896",
  "fulltext_url": null,
  "id": "f8g38f07a185c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09896",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7212129090",
  "pdf_url": "https://arxiv.org/pdf/2609.09896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f8g38f07a185c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs",
  "title_text": "Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/f8g38f07a185c",
  "venue": "arXiv",
  "versions": []
}