{
  "abstract": "We study an optimization problem for weighted Lin--Lu--Yau curvature. Given a graph $G=(V,E)$ and a prescribed lower curvature bound $k\\ge0$, we ask for the smallest multiplicative distortion \\[ \\Dist(w):=\\frac{\\sup_{e\\in E}w_e}{\\inf_{e\\in E}w_e} \\] of the edge weights needed to make every edge have curvature at least $k$. This leads to the family \\[ \\DN_{\\LLY}^{k}(G) := \\inf\\bigl\\{ \\Dist(w): κ_{\\LLY}^{w}(e)\\ge k \\text{ for every }e\\in E(G) \\bigr\\}, \\] which extends the nonnegative-curvature invariant introduced in our preceding paper \\cite{Xia2026InvariantI}. The edge weights change the transition probabilities, while the combinatorial graph distance remains fixed. Positive prescribed curvature leads to a fundamental distinction from the case $k=0$. If $G$ is a connected locally finite infinite graph and $k>0$, then \\[ \\DN_{\\LLY}^{k}(G)=\\infty. \\] Thus no finite multiplicative distortion of the edge weights can impose a uniform positive curvature lower bound on such a graph. For a finite tree $T$ with $m$ edges, such weights exist exactly when $k\\le2/m$. We prove that the minimum distortion is attained by a weight unique up to scaling, and that the normalized minimizer and its distortion depend continuously and nondecreasingly on $k$. We then extend the finite-tree theory to general connected graphs. As an application, we characterize the graphs of girth at least six for which finite $k$-distortion is possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21454",
  "author_total": 1,
  "authors": [
    {
      "name": "Qing Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mbet0njcnz4w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21454",
  "fulltext_url": null,
  "id": "9mbet0njcnz4w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21454",
  "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/2610.21454",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mbet0njcnz4w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21454",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21454"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds",
  "title_text": "Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/9mbet0njcnz4w",
  "venue": "arXiv",
  "versions": []
}