{
  "abstract": "The booksize $\\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollobás and Erdős, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502--520] asked whether the same conclusion holds under the spectral condition $λ(G)>λ(T_{n,2})$, where $λ(G)$ is the spectral radius of the adjacency matrix. We answer this question in a strong form: every $n$-vertex graph $G\\neq T_{n,2}$ with $λ(G)\\geλ(T_{n,2})$ satisfies \\[ \\mathrm{bk}(G)\\ge\\max\\Big\\{\\frac13λ(G),\\,λ(G)-\\frac n3,\\,2λ(G)-n\\Big\\}. \\] Consequently, the condition $λ(G)>λ(T_{n,2})$ forces $\\mathrm{bk}(G)\\ge\\lfloor n/6\\rfloor+1$. The middle term is a spectral improvement of Edwards' bound $\\mathrm{bk}(G)\\ge\\frac{2m}{n}-\\frac n3$, and all three bounds are best possible. These results come from the edge-spectral setting: every graph $G$ with $m$ edges and $λ(G)\\ge\\sqrt m$ that is not a complete bipartite graph satisfies \\[ \\mathrm{bk}(G)\\ge\\max\\Big\\{λ(G)-\\frac{2m}{3λ(G)},\\,2λ(G)-\\frac{2m}{λ(G)}\\Big\\}, \\] which strengthens the bound $\\mathrm{bk}(G)\\ge\\frac13λ(G)$ of Zhao, You, Zeng and Zhang. As an application of our method, we prove a triangle counting bound $t(G)\\ge\\frac13(λ(G)+1)(λ(G)^2-m)$, which improves the result of Bollobás and Nikiforov [J. Combin. Theory Ser B. (2007)]. Finally, we prove stability results at both thresholds: if $λ(G)\\ge(\\frac12-o(1))n$, then either $\\mathrm{bk}(G)\\ge(\\frac16-o(1))n$ or $G$ can be made into $T_{n,2}$ by adding and deleting $o(n^2)$ edges; if $λ(G)\\ge(1-o(1))\\sqrt m$, then either $\\mathrm{bk}(G)\\ge(\\frac13-o(1))\\sqrt m$ or $G$ differs from a complete bipartite graph in $o(m)$ edges.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20225",
  "author_total": 3,
  "authors": [
    {
      "name": "Yongtao Li"
    },
    {
      "name": "Lele Liu"
    },
    {
      "name": "Bo Ning"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckfyd90fryjvd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20225",
  "fulltext_url": null,
  "id": "ckfyd90fryjvd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20225",
  "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/2609.20225",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckfyd90fryjvd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.20225",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20225"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The spectral Erdős book theorem: sharp bounds and stability",
  "title_text": "The spectral Erdős book theorem: sharp bounds and stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/ckfyd90fryjvd",
  "venue": "arXiv",
  "versions": []
}