{
  "abstract": "A graph $G$ with $m$ edges is a Nosal graph if its spectral radius $λ(G)$ exceeds $\\sqrt m$. At the threshold $λ(G)=\\sqrt m$, the star $K_{1,m}$ shows that one vertex can be incident with all $m$ edges. In contrast, Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026)] proved that every Nosal graph $G$ with sufficiently many edges satisfies $Δ(G)\\le\\frac12m+m^{0.99}$, where the constant $\\frac12$ is best possible. In this paper, we prove that every Nosal graph $G$ with spectral radius $λ$ satisfies $Δ(G)<m-\\frac12(λ-1)(λ-2)$, and hence $Δ(G)<\\frac12m+\\frac32\\sqrt m-1$. Both bounds are sharp up to an additive constant less than $\\frac12$: for every $r\\ge3$, the complete graph $K_r$ with $\\frac12(r^2+r-2)$ pendant vertices attached to one vertex is a Nosal graph with $r^2-1$ edges whose maximum degree lies within $\\frac12$ of both bounds. Consequently, the largest maximum degree of a Nosal graph with $r^2-1$ edges is exactly $\\frac12(r^2+3r-4)$. The first bound cannot be improved by any positive additive constant: its gap on these graphs is less than $\\frac{4}{r+5}$ and tends to zero as $r\\to \\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40380",
  "author_total": 1,
  "authors": [
    {
      "name": "Yongtao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpy00bqvtyqnd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40380",
  "fulltext_url": null,
  "id": "fpy00bqvtyqnd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40380",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40380",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpy00bqvtyqnd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40380",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40380"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The maximum degree of Nosal graphs",
  "title_text": "The maximum degree of Nosal graphs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:51Z",
  "url": "https://research.draconitic.com/paper/fpy00bqvtyqnd",
  "venue": "arXiv",
  "versions": []
}