图的曲率畸变数：非负 Lin--Lu--Yau 曲率
Curvature-Distortion Numbers of Graphs: Nonnegative Lin--Lu--Yau Curvature
- University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出图的曲率畸变数，度量使 Lin--Lu--Yau 曲率非负所需的最小边权展宽，证明其限制树的分支数，并给出无限树及高围长图的有限畸变分类。
AI中文摘要：
我们引入了曲率畸变数，这是一个尺度不变参数，用于度量为使加权离散曲率处处非负所需的正边权的最小乘法展宽。我们在传输距离为固定的组合图距离时，发展了 Lin--Lu--Yau 曲率的理论。对于树，该不变量具有显式的非线性不动点描述，这产生了一个在缩放意义下唯一的最优权重。更重要的是，曲率畸变数控制了树的分支拓扑：对于每棵有限树 $T$，\\[ |B(T)|\le \left\lceil \DN_{\LLY}(T)\right\rceil, \\] 其中 $B(T)$ 是分支顶点集。因此，为实现非负曲率所需的权重畸变量对树的拓扑复杂度施加了直接的定量限制。对于局部有限的无限树，有限畸变被完全分类：它恰好发生在双射线以及由有限树通过附加单条射线得到的一端树上。曲率畸变与树拓扑之间的这种联系通过由不在长度为 $3$、$4$ 或 $5$ 的环中的边构成的子图自然地推广到一般连通图。每当曲率畸变数有限时，除非整个图是长度至少为 $6$ 的环，否则这个类树部分是森林，并且其每个树分量继承了相应的畸变和拓扑界限。特别地，树理论为围长至少为 $6$ 的图提供了完整的有限畸变分类。
英文摘要：
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 $6$.
