Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds
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≥0, we ask for the smallest multiplicative distortion Dist(w):=(supe∈ Ewe)/(infe∈ Ewe) of the edge weights needed to make every edge have curvature at least k. This leads to the family DNLLYk(G) := inf{ Dist(w): κLLYw(e)≥ k for every e∈ E(G) }, 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 DNLLYk(G)=∞. 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≤2/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.
Related
- Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature
- Upper bounds for ordered Ramsey numbers of forests and bounded-degree graphs
- Macroscopic scalar curvature bounds on graphs
- Counterexamples to prescribed scalar curvature in the null case
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
BibTeX
@misc{xia2026curvaturedistortion,
title = {{Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds}},
author = {Qing Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21454},
eprint = {2610.21454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21454},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds
AU - Qing Xia
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21454
UR - https://arxiv.org/abs/2610.21454
AB - We study an optimization problem for weighted Lin--Lu--Yau curvature. Given a graph G=(V,E) and a prescribed lower curvature bound k≥0, we ask for the smallest multiplicative distortion Dist(w):=(sup_e∈ Ew_e)/(inf_e∈ Ew_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{ Dist(w): κ_LLY^w(e)≥ k for every e∈ E(G) }, 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)=∞. 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≤2/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.
ER -