{
  "abstract": "We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13551",
  "author_total": 3,
  "authors": [
    {
      "name": "Cecelia Higgins"
    },
    {
      "name": "Pieter Spaas"
    },
    {
      "name": "Sofia Torelli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8hmegrscf47wv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13551",
  "fulltext_url": null,
  "id": "8hmegrscf47wv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13551",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.LO"
    ],
    [
      "arxiv",
      "math.OA"
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      "arxiv",
      "math.SP"
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      "openalex",
      "2607|Limits and Structures in Graph Theory"
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  "openalex": "W7221525339",
  "pdf_url": "https://arxiv.org/pdf/2610.13551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8hmegrscf47wv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs",
  "title_text": "Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/8hmegrscf47wv",
  "venue": "arXiv",
  "versions": []
}