{
  "abstract": "Let $ρ(n,d)$ denote the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $d$. A graph attaining this maximum is called an extremal graph. Liu [J. Combin. Theory Ser. B, 2024] determined the extremal graphs for $d=3,4$ and formulated two conjectures for general $d$. For each fixed odd integer $d\\ge3$, the conjectures assert that: (1) $\\displaystyle\\lim_{n\\to\\infty}n^2\\bigl(d-ρ(n,d)\\bigr) =(d-1)π^2/4$. (2) For all sufficiently large $n$, the degree sequence of every extremal graph is $(d,\\ldots,d,d-1)$ for odd $n$ and $(d,\\ldots,d,1)$ for even $n$. We prove the first conjecture for every fixed odd $d\\ge3$ and, more precisely, obtain the asymptotic expansion \\[ ρ(n,d) =d-\\frac{(d-1)π^2}{4n^2} +\\frac{(d-1)^2π^2}{4n^3} +O_d(n^{-4}) \\qquad(n\\to\\infty). \\] We further prove the second conjecture for every fixed odd $d\\ge3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28706",
  "author_total": 3,
  "authors": [
    {
      "name": "Liangdong Fan"
    },
    {
      "name": "Liying Kang"
    },
    {
      "name": "Yaojun Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1422f3vhjdx14.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28706",
  "fulltext_url": null,
  "id": "1422f3vhjdx14",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28706",
  "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": "W7214313089",
  "pdf_url": "https://arxiv.org/pdf/2609.28706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1422f3vhjdx14.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nonregular graphs of odd maximum degree with maximum spectral radius",
  "title_text": "Nonregular graphs of odd maximum degree with maximum spectral radius",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1422f3vhjdx14",
  "venue": "arXiv",
  "versions": []
}