{
  "abstract": "We introduce the curvature-distortion number, a scale-invariant parameter measuring the least multiplicative spread of positive edge weights required to make a weighted discrete curvature nonnegative everywhere. We develop the theory for Lin--Lu--Yau curvature when the transport distance is the fixed combinatorial graph distance. For trees, the invariant admits an explicit nonlinear fixed-point description, which produces a canonical optimal weight that is unique up to scaling. More importantly, the curvature-distortion number controls the branching topology of the tree: for every finite tree $T$, \\[ |B(T)|\\le \\left\\lceil \\DN_{\\LLY}(T)\\right\\rceil, \\] where $B(T)$ is the set of branch vertices. Thus the amount of weight distortion required to achieve nonnegative curvature imposes a direct quantitative restriction on the topological complexity of the tree. For locally finite infinite trees, finite distortion is classified completely: it occurs precisely for the double ray and for one-ended trees obtained from a finite tree by attaching a single ray. This connection between curvature distortion and tree topology extends naturally to general connected graphs through the subgraph formed by edges lying in no cycle of length $3$, $4$, or $5$. Whenever the curvature-distortion number is finite, this tree-like part is a forest unless the whole graph is a cycle of length at least $6$, and each of its tree components inherits the corresponding distortion and topological bounds. In particular, the tree theory yields complete finite-distortion classifications for graphs of girth at least $6$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12125",
  "author_total": 1,
  "authors": [
    {
      "name": "Qing Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hf9yrfnsmb6x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12125",
  "fulltext_url": null,
  "id": "2hf9yrfnsmb6x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12125",
  "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": "W7212508671",
  "pdf_url": "https://arxiv.org/pdf/2609.12125",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hf9yrfnsmb6x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.12125",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12125"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature",
  "title_text": "Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/2hf9yrfnsmb6x",
  "venue": "arXiv",
  "versions": []
}