Eigenvalue ratios of nonnegatively curved graphs

TL;DR

This paper establishes an optimal eigenvalue ratio estimate for finite weighted graphs with nonnegative curvature, providing a new method for spectral analysis and insights into graph construction and expanders.

Contribution

It introduces a size-independent eigenvalue ratio estimate for graphs satisfying $CD(0, olinebreak\infty)$ and demonstrates how Cartesian products can generate such graphs.

Findings

Optimal eigenvalue ratio estimate derived

Cartesian products effectively construct $CD(0, olinebreak\\infty)$ graphs

Higher order Cheeger constant ratio estimates discussed

Abstract

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality $CD(0,\infty)$. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfying $CD(0,\infty)$. We also discuss a higher order Cheeger constant ratio estimate and related topics about expanders.