{
  "abstract": "We develop a variational framework for extremal Lin--Lu--Yau curvature under positive edge reweighting with the combinatorial metric fixed. For a connected locally finite graph $G$, define \\[ \\Kmax(G) := \\sup_{w>0}\\inf_{e\\in E(G)}κ_{\\LLY}^w(e). \\] This invariant relates edgewise discrete curvature to global combinatorial and topological data. In the high-girth regime, we prove for every connected locally finite graph the exact identity \\[ \\Kmax(G)=\\frac4{\\mad(G)}-2, \\] with the convention $1/\\infty=0$. The formula is also equivalent to a normalized Euler-characteristic density: \\[ \\Kmax(G) = \\inf_{\\substack{H\\subseteq G\\text{ finite}\\\\E(H)\\ne\\varnothing}} \\frac{2(c(H)-β(H))}{|E(H)|}. \\] For arbitrary locally finite graphs, a nonnegative short-cycle surplus yields girth-dependent density bounds, with rigidity results in the finite case. We also classify the locally finite graphs of girth at least $6$ admitting a positive edge weight with nonnegative curvature on every edge: they are precisely the trees and the finite cycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08670",
  "author_total": 1,
  "authors": [
    {
      "name": "Qing Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3w05ye8azkg0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08670",
  "fulltext_url": null,
  "id": "e3w05ye8azkg0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08670",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212013099",
  "pdf_url": "https://arxiv.org/pdf/2609.08670",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3w05ye8azkg0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.08670",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08670"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles",
  "title_text": "Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/e3w05ye8azkg0",
  "venue": "arXiv",
  "versions": []
}