Discrete quantitative nodal theorem

TL;DR

This paper presents a generalized theorem linking eigenvalues of graph Laplacians to properties of subgraphs, showing that certain spectral gaps imply strong connectivity and expansion properties of associated subgraphs.

Contribution

It introduces a unified theorem that extends the Discrete Nodal Theorem and Cheeger's Inequality, relating spectral gaps to subgraph connectivity and expansion.

Findings

Eigenvalue gaps imply connected, edge-expanding subgraphs

Subgraphs induced by eigenvector supports are connected and expand

Generalizes classical spectral graph theorems

Abstract

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. special case of this result will assert that if the second and third eigenvalues of the Laplacian are at least c apart, then the subgraphs induced by the positive and negative supports of the eigenvector belonging to the second eigenvalue are not only connected, but edge-expanders (in a weighted sense, with expansion depending on c).