{
  "abstract": "Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $Γ_E(R)$ and the associate class graph $Γ_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $Γ(R)$, $Γ_A(R)$ and $Γ_E(R)$. We prove that the count for $Γ_A(R)$ is bounded below by the count for $Γ_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $Γ_A(R)\\congΓ_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $Γ(R)\\congΓ_E(R)$ if and only if $R\\cong \\mathbb Z_2^{\\,n}$ for some $n\\ge2$, or $R\\cong\\mathbb Z_4$, or $R\\cong\\mathbb Z_2[x]/(x^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36021",
  "author_total": 2,
  "authors": [
    {
      "name": "Ganesh S. Kadu"
    },
    {
      "name": "Vishnu Tanpure"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/925xygm4rq0ts.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36021",
  "fulltext_url": null,
  "id": "925xygm4rq0ts",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36021",
  "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.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214896531",
  "pdf_url": "https://arxiv.org/pdf/2609.36021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/925xygm4rq0ts.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Zero divisors of Gorenstein Rings",
  "title_text": "Zero divisors of Gorenstein Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/925xygm4rq0ts",
  "venue": "arXiv",
  "versions": []
}