{
  "abstract": "For an odd prime $p$, let $Γ(R)$ denote the zero-divisor graph associated with the finite local ring $R= \\mathbb{F}_p[x]/\\langle x^4 \\rangle$, having $p^3 - 1$ vertices and size $|E(Γ(R))| = \\frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$. Using an equitable 4-cell partition induced by the ideal filtration $\\langle u^3 \\rangle \\subset \\langle u^2 \\rangle \\subset \\langle u \\rangle$, we prove that $\\operatorname{Spec}_A(Γ(R))$ decomposes into discrete levels $λ\\in \\{0, -1\\}$ with multiplicities $p^3 - p^2 - 1$ and $p^2 - 3$, respectively, alongside three irrational roots of an irreducible cubic polynomial $P_3(λ) = 0$, yielding adjacency energy $ε(Γ(R)) = 4p^2 + \\mathcal{O}(p)$. In contrast, the Laplacian spectrum is entirely integral: $\\operatorname{Spec}_L(Γ(R)) = \\{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\\}$. Consequently, the tree entropy satisfies $z(Γ(R)) = \\ln p - \\frac{1}{p} + \\frac{\\ln p}{p} + \\mathcal{O}(\\frac{\\ln p}{p^2})$, with small-failure unreliability bounded by $p^2(p - 1)(1 - q)^{p-1}$. In linear consensus $\\dot{x} = -Lx$, error trajectories decouple into three modal timescales $\\{ (p^3 - 1)^{-1}, (p^2 - 1)^{-1}, (p - 1)^{-1} \\}$ with algebraic connectivity $α(Γ(R)) = p - 1$ and steady-state $H_2$ error variance $H_2^2 = \\frac{Kf(Γ(R))}{2n^2} = \\frac{1}{2p} + \\mathcal{O}(p^{-3})$. Furthermore, the Kirchhoff index yields a vanishing average resistance distance $\\overline{r}(Γ(R)) = \\frac{2}{p} + \\mathcal{O}(p^{-3}) \\to 0$, while unbiased Markovian random walks induce an asymptotic traffic concentration vector $Π= (\\frac{1}{3}, 0, \\frac{1}{3}, \\frac{1}{3})$ with quadratic per-node core routing stress $π_{\\mathrm{core}}/π_{\\mathrm{periph}} \\sim p^2 + p + 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22909",
  "author_total": 3,
  "authors": [
    {
      "name": "Apurba Sarkar"
    },
    {
      "name": "Kalyan Hansda"
    },
    {
      "name": "Makhan Maji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70k3fwakjx2d2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22909",
  "fulltext_url": null,
  "id": "70k3fwakjx2d2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22909",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7214029623",
  "pdf_url": "https://arxiv.org/pdf/2609.22909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70k3fwakjx2d2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Graph theory and applications",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $\\mathbb{F}_p[x]/\\langle x^4\\rangle$: Consensus, Reliability, and Transport",
  "title_text": "Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over 𝔽_p[x]/⟨ x^4⟩: Consensus, Reliability, and Transport",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/70k3fwakjx2d2",
  "venue": "arXiv",
  "versions": []
}