发表机构
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 中文总结
本文研究正边权下 Lin--Lu--Yau 曲率的一致下界极值问题，对围长至少 6 的图精确给出极值曲率公式，并刻画短圈贡献与取等刚性。
AI 中文摘要
我们考虑极值曲率问题，即在正边权上优化一致离散曲率下界，并在此针对 Lin--Lu--Yau 曲率发展该问题。设 $G=(V,E)$ 为有限连通图，$w:E\to(0,\infty)$ 为正边权。在固定组合距离的加权 Lin--Lu--Yau 模型中，记 \\[ \kappa_{\LLY}^w(G):=\min_{e\in E}\kappa_{\LLY}^w(e) \\] 并定义极值 Lin--Lu--Yau 曲率 \\[ \Kmax(G):=\sup_{w>0}\kappa_{\LLY}^w(G). \\] 对于围长至少为 $6$ 的图，我们精确确定该不变量：\\[ \Kmax(G)=\frac{4}{\mad(G)}-2, \\] 其中 $\mad(G)$ 是最大平均度。等价地，\\[ \Kmax(G) =\min_{\substack{H\subseteq G\text{ connected}\E(H)\ne\varnothing}} \frac{2(1-\beta(H))}{|E(H)|}, \\] 其中 $\beta(H)=|E(H)|-|V(H)|+1$ 是连通图 $H$ 的圈秩。因此，在高围长情形下，该不变量是归一化的欧拉示性数密度。我们以经典严格平衡概念刻画取等条件，并证明当上确界未取到时，最大化序列在精确归一化关联意义下集中于适当的密度最大核。对于任意有限图，我们通过非负盈余分离短圈的贡献，该盈余在恰不包含于长度为 $3$、$4$ 或 $5$ 的圈的边上为零。对于不包含于任何三角形的边，该盈余是显式局部分数匹配问题的值。这产生尖锐层级 \\[ \Kmax(G)\le 4-\ell+\frac{\ell-2}{\mad(G)}, \qquad \girth(G)\ge\ell,\quad \ell\in\{3,4,5,6\}, \\] 当 $\ell=6$ 时对每个有限连通图取等。我们还证明取等是刚性的。
英文摘要
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 \[ \Kmax(G) := \sup_{w>0}\inf_{e\in E(G)}κ_{\LLY}^w(e). \] 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 \[ \Kmax(G)=\frac4{\mad(G)}-2, \] with the convention $1/\infty=0$. The formula is also equivalent to a normalized Euler-characteristic density: \[ \Kmax(G) = \inf_{\substack{H\subseteq G\text{ finite}\\E(H)\ne\varnothing}} \frac{2(c(H)-β(H))}{|E(H)|}. \] 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.
Comments29pages,Revised version. The high-girth extremal-curvature formula is extended from finite graphs to connected locally finite graphs. New results on infinite graphs and the nonnegative-curvature classification are added, and the exposition has been substantially revised
