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PreprintCC BY 4.0arXiv

Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles

  • Qing Xia
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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) := supw>0infe∈ E(G)κLLYw(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)=4/mad(G)−2, with the convention 1/∞=0. The formula is also equivalent to a normalized Euler-characteristic density: Kmax(G) = infsubstackH⊆ G finite E(H)≠∅ (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.

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Cite

BibTeXRIS
BibTeX
@misc{xia2026extremal,
  title = {{Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles}},
  author = {Qing Xia},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.08670},
  eprint = {2609.08670},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08670},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles
AU  - Qing Xia
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.08670
UR  - https://arxiv.org/abs/2609.08670
AB  - 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>0inf_e∈ 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)=4/mad(G)−2, with the convention 1/∞=0. The formula is also equivalent to a normalized Euler-characteristic density: Kmax(G) = inf_substackH⊆ G finite E(H)≠∅ (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.
ER  -