{
  "abstract": "We extend the Bakry--Émery curvature framework on graphs, broadening its applications and strengthening the resulting diameter bounds. We study diameter bounds and finiteness for amply regular graphs, distance-regular graphs, $(s,c,a,k)$-graphs, and locally strongly regular graphs. In particular, for an amply regular graph with parameters $(v,k,λ,μ)$, we prove that \\[{\\rm{diam}}(G)\\le4+8\\log\\frac{k}{λ+2}\\] when $μ>2$ and $k\\ge3λ^2/μ$, improving the previous linear bound $4k/(λ+4)$ to a logarithmic bound. We also address several related conjectures. In their monograph \\emph{Distance-Regular Graphs}, Brouwer, Cohen, and Neumaier conjectured that every amply regular graph with $μ>1$, except the icosahedron, satisfies ${\\rm{diam}}(G)\\le2k/(λ+2)$. We confirm this bound when $μ>λ+1$ and give counterexamples to the conjecture with $μ=λ$. For amply regular graphs with $μ\\geλ$ and $μ>1$, we obtain ${\\rm{diam}}(G)\\le 2k/μ+1$. This proves the Qiao--Park--Koolen conjecture. In 1983, Terwilliger conjectured that every $(s,c,a,k)$-graph with $c>1$ is finite. We prove this conjecture when $s>2$, $c>2$, and $(s,c)\\ne(4,3)$. Moreover, we prove Weetman's finiteness conjecture for locally strongly regular graphs when the local parameter $μ>1$, and disprove his stronger conjecture that every strongly regular graph is an $\\mathscr{S}$-graph. We also prove that every graph whose local graphs all have rank 3 is finite, which in particular proves a conjecture of Soicher in 1992.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36620",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaizhe Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1w993snn3agdr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36620",
  "fulltext_url": null,
  "id": "1w993snn3agdr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36620",
  "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|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7224130416",
  "pdf_url": "https://arxiv.org/pdf/2610.36620",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1w993snn3agdr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36620",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36620"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Diameter Bounds and Finiteness of Graphs from Local Intersection Data",
  "title_text": "Diameter Bounds and Finiteness of Graphs from Local Intersection Data",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1w993snn3agdr",
  "venue": "arXiv",
  "versions": []
}