Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature
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)|≤ ⌈ DNLLY(T)⌉, 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 $6$.
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BibTeX
@misc{xia2026curvaturedistortion,
title = {{Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature}},
author = {Qing Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12125},
eprint = {2609.12125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12125},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature
AU - Qing Xia
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.12125
UR - https://arxiv.org/abs/2609.12125
AB - 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)|≤ ⌈ DN_LLY(T)⌉, 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 $6$.
ER -