{
  "abstract": "We prove a sharp quadratic inequality between the largest two eigenvalues $λ_1 \\ge λ_2$ of a graph with $n$ vertices. We also prove a quadratic inequality between the second and third largest eigenvalues $λ_2 \\ge λ_3$. These results in particular imply the bounds $λ_1 + λ_2 \\le \\frac{8}{7} n - 2$, $λ_3 \\le \\frac{n}{3} - 1$ and $λ_2 + λ_3 \\le \\frac{2}{3} n - 2$. In fact we determine the closure of the set of possible $(\\frac{λ_1+1}{n}, \\frac{λ_2+1}{n}) \\in \\mathbb{R}^2$ and $(\\frac{λ_2+1}{n}, \\frac{λ_3+1}{n}) \\in \\mathbb{R}^2$. More generally, we prove quadratic bounds in the case of symmetric matrices in $[0,1]^{n \\times n}$, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in $[-1,1]^{n \\times n}$. These bounds are proved by transforming the problem into extremal geometric questions in $\\mathbb{R}^3$ and $\\mathbb{R}^2$. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37980",
  "author_total": 1,
  "authors": [
    {
      "name": "Roland Paulin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0h0k7836pjy37.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37980",
  "fulltext_url": null,
  "id": "0h0k7836pjy37",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37980",
  "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"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7214931116",
  "pdf_url": "https://arxiv.org/pdf/2609.37980",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0h0k7836pjy37.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37980",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37980"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Quadratic inequalities between the largest eigenvalues of a graph",
  "title_text": "Quadratic inequalities between the largest eigenvalues of a graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0h0k7836pjy37",
  "venue": "arXiv",
  "versions": []
}