{
  "abstract": "Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least $g$ and the standard second-moment bound for graph spectra. We prove that the supremum of $(λ_1+λ_n)/n$ over these lists lies between $(1-o(1))(\\log g)^2/g^3$ and $(10/7+o(1))(\\log g)^2/g^3$ as $g\\to\\infty$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\\geq5$, we obtain a stronger bound $(λ_1+λ_n)/n \\leq 12/((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05172",
  "author_total": 1,
  "authors": [
    {
      "name": "Chen Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/djj8jj422q4ha.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05172",
  "fulltext_url": null,
  "id": "djj8jj422q4ha",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05172",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7220695074",
  "pdf_url": "https://arxiv.org/pdf/2610.05172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/djj8jj422q4ha.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotic Spectral Bounds under Odd-Girth Constraints",
  "title_text": "Asymptotic Spectral Bounds under Odd-Girth Constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/djj8jj422q4ha",
  "venue": "arXiv",
  "versions": []
}