{
  "abstract": "For a graph \\(F\\), the Turán number \\(\\ex(n,F)\\) is the maximum number of edges in an \\(F\\)-free graph on \\(n\\) vertices. Let \\(q\\ge 2\\) be an integer and set \\(n=q^{2}+q+1\\). Brown and Erdős, Rényi and Sós independently proved that $\\ex(n,C_{4})\\ge \\frac12 q(q+1)^{2}$ for every prime power \\(q\\), and Füredi subsequently established the upper bound $\\frac12 q(q+1)^{2}$ for $\\ex(n,C_{4})$ whenever \\(q\\notin\\{1, 7,9,11,13\\}\\). In this article, we prove that every \\(C_{4}\\)-free graph \\(G\\) on \\(n\\) vertices with at most \\(\\frac12 q(q+1)^{2}\\) edges satisfies $τ(G)\\le n^{(n-3)/2}$, where \\(τ(G)\\) denotes the number of spanning trees of \\(G\\). In particular, for every prime power $q\\notin\\{7,9,11,13\\}$, the above upper bound on $τ(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34616",
  "author_total": 3,
  "authors": [
    {
      "name": "Shaohan Xu"
    },
    {
      "name": "Fengming Dong"
    },
    {
      "name": "Kexiang Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4tn8n8fqq9kg3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34616",
  "fulltext_url": null,
  "id": "4tn8n8fqq9kg3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34616",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214771516",
  "pdf_url": "https://arxiv.org/pdf/2609.34616",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4tn8n8fqq9kg3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34616",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34616"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Turán-type extremal problem for the number of spanning trees in $C_4$-free graphs",
  "title_text": "A Turán-type extremal problem for the number of spanning trees in C_4-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4tn8n8fqq9kg3",
  "venue": "arXiv",
  "versions": []
}