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
该研究将图的边连通性与Lin-Lu-Yau曲率的关系推广到超图，证明了满足非负曲率的局部有限连通r-均匀线性超图的边连通性等于其最小关联度，且非线性超图中二者差距可任意大。
AI 中文摘要
Chen、Liu和You[1]证明了，具有正Lin-Lu-Yau曲率的局部有限连通图，其边连通性等于其最小度。Liu和Xia[2]随后表明，对于每个具有非负Lin-Lu-Yau曲率的有限连通图，该结论同样成立，并对所有无限例外情况进行了分类。我们研究了Tian和Zhao[3]引入的超图随机游走曲率对应的问题。我们构建了Liu和Xia[2]所用组合不等式的超图类似物，并利用它研究均匀线性超图中的边割。我们的第一个主要结果指出，每个满足r≥3的局部有限连通r-均匀线性超图，若具有非负Lin-Lu-Yau曲率，其边连通性等于其最小关联度。线性假设至关重要。特别地，对于每个r≥3和每个整数t≥2，我们构造了一个有限连通简单非线性r-均匀超图，其具有正Lin-Lu-Yau曲率，但边连通性比其最小度小t。因此，即使在严格正曲率下，非线性情形中最小度与边连通性之间的差距也可以任意大。
英文摘要
We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected $r$-uniform linear hypergraph with $r\ge3$ and nonnegative Lin--Lu--Yau curvature has this property. Among the various extensions of Ollivier--Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval $[1/2,1]$. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.
Comments30 pages. Major revision incorporating the idleness-function results of arXiv:2608.01970 into a unified paper. Corrected the attribution and discussion of the Tian-Zhao random walk and clarified its relation to the ORCHID equal-edges walk. Several citation inaccuracies were also corrected. Main results unchanged
