Gaps between consecutive eigenvalues for compact metric graphs

TL;DR

This paper investigates the eigenvalue gaps of the Laplacian on compact metric graphs, providing new bounds and extensions that depend on graph length and edge properties, advancing spectral graph theory understanding.

Contribution

It introduces a Cheeger-type lower bound on the eigenvalue gap and improves known upper bounds for metric trees and other graph types.

Findings

Established a Cheeger-type lower bound on eigenvalue gaps.

Improved upper bounds for eigenvalue ratios on metric trees.

Extended bounds to certain other classes of graphs.

Abstract

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends only on the total length of the graph and minimum edge length. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees and extensions to certain other types of graphs.