标题： 组合Courant-Fischer-Weyl极小极大原理在加权树的Cheeger$k$常数上的应用 显示英文标题

标题： Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests

摘要： 我们建立了加权森林中Cheeger$k$-常数的新极小极大和极大极小刻画，从而提供了Courant-Fischer-Weyl极大极小原理的第一个组合类比。 就应用而言，我们证明了森林1-Laplacian变分特征值与典型指标的选择无关；我们提出了一个改进的高阶Cheeger不等式，涉及图的环数和$p$-Laplacian特征值；我们给出了一个组合证明，证明了等式$h_k=\lambda_k(\Delta_1)$，该等式将1-Laplacian变分特征值与多路Cheeger常数联系起来。 显示英文摘要

摘要： We establish novel max-min and minimax characterizations of Cheeger $k$-constants in weighted forests, thereby providing the first combinatorial analogue of the Courant-Fischer-Weyl minimax principle. As for applications, we prove that the forest 1-Laplacian variational eigenvalues are independent of the choice of typical indexes; we propose a refined higher order Cheeger inequality involving numbers of loops of graphs and $p$-Laplacian eigenvalues; and we present a combinatorial proof for the equality $h_k=\lambda_k(\Delta_1)$ which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants.