Neumann Cheeger constants on graphs

TL;DR

This paper introduces Cheeger constants for Neumann Laplacian problems on graphs, inspired by Riemannian geometry, and establishes estimates relating these constants to the first nontrivial eigenvalues.

Contribution

It defines Neumann Cheeger constants on graphs and proves Cheeger inequalities connecting these constants to eigenvalues, extending geometric analysis to graph Laplacians.

Findings

Defined Neumann Cheeger constants for graphs

Proved Cheeger inequalities for Neumann eigenvalues

Extended Riemannian geometric concepts to graph theory

Abstract

For any subgraph of a graph, the Laplacian with Neumann boundary condition was introduced by Chung and Yau [CY94]. In this paper, motivated by the Riemannian case, we introduce the Cheeger constants for Neumann problems and prove corresponding Cheeger estimates for first nontrivial eigenvalues.