发表机构
Indian Institute of Technology Delhi（印度德里理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为非均匀超图建立谱框架，证明 Cheeger 不等式，并构造对 Alon--Boppana 界紧的最优超图扩展器族。
AI 中文摘要
超图为建模高阶关系提供了自然框架，但为一般非均匀超图开发具有可证明保证的谱技术仍然具有挑战性。基于 Banerjee 的归一化邻接矩阵和 Spiro 的基于平均的扩散框架，我们为非均匀超图开发了一个谱框架，并建立了其电导的 Cheeger 不等式。超图谱理论中的一个基本结果断言，对于每个非覆盖超图，其归一化拉普拉斯算子的第二小特征值至多为 1。这一谱刻画产生了针对非覆盖超图的改进的 Cheeger 不等式，并且我们使用循环超图和立方体超图证明了所得不等式在两侧都是紧的。我们的框架进一步产生了高阶 Cheeger 不等式，并为 Fiedler 谱划分算法提供了理论保证，所有这些都在超图的设置中。最后且最值得注意的是，我们构造了一个新的最优超图扩展器族，该族对 Alon--Boppana 界是紧的。
英文摘要
Hypergraphs provide a natural framework for modeling higher-order relationships, but the development of spectral techniques with provable guarantees for general non-uniform hypergraphs remains challenging. Building on Banerjee's normalized adjacency matrix and Spiro's averaging-based diffusion framework, we develop a spectral framework for non-uniform hypergraphs and establish Cheeger's inequality for their conductance. A fundamental result in the spectral theory of hypergraphs asserts that, for every non-covering hypergraph, the second-smallest eigenvalue of its normalized Laplacian is at most one. This spectral characterization yields an improved Cheeger's inequality for non-covering hypergraphs, and we show that the resulting inequality is tight on both sides using cycle and cube hypergraphs. Our framework further yields higher-order Cheeger inequalities and provides theoretical guarantees for Fiedler's spectral partitioning algorithm, all in the setting of hypergraphs. Finally and most notably, we construct a new family of optimal hypergraph expanders that is tight for the Alon--Boppana bound.
