{
  "abstract": "Let $K$ be a finite pure ridge-connected $d$-dimensional simplicial complex with at least two facets, and let $m_K(λ)$ denote the multiplicity of a positive eigenvalue $λ$ of its upper Laplacian $L_{d-1}^{\\mathrm{up}}(K)$. We obtain upper bounds for $m_K(λ)$ in terms of the basic combinatorial parameters $f_d(K)$ and $f_{d-1}(K)$, the numbers of facets and ridges, $p_d(K)$, the number of pendant facets, and $q_d(K)$, the number of quasi-pendant ridges. The quantity $θ_d(K)=d f_d(K)-f_{d-1}(K)+1$ arises as the cyclomatic number of a natural reduced facet--ridge incidence graph. We prove \\[ m_K(λ)\\le 2θ_d(K)+p_d(K), \\qquad m_K(λ)\\le 2θ_d(K)+q_d(K)\\quad(λ\\ne d). \\] We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure $d$-dimensional simplicial complex, we prove the sharp inequality $m_K(d)\\ge p_d(K)-q_d(K)$, which is a higher-dimensional analogue of Faria's inequality. For $d=1$, our results reduce to the corresponding multiplicity bounds for graph Laplacians.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29677",
  "author_total": 1,
  "authors": [
    {
      "name": "Vinayak Gupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dhg73914sgnqn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29677",
  "fulltext_url": null,
  "id": "dhg73914sgnqn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29677",
  "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": "W7214341845",
  "pdf_url": "https://arxiv.org/pdf/2609.29677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dhg73914sgnqn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes",
  "title_text": "Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/dhg73914sgnqn",
  "venue": "arXiv",
  "versions": []
}