Spectral Estimates for Infinite Quantum Graphs

TL;DR

This paper studies the spectral properties of infinite quantum graphs, introducing a combinatorial isoperimetric constant and establishing Cheeger-type estimates to analyze their spectra.

Contribution

It introduces a new combinatorial isoperimetric constant for quantum graphs and proves Cheeger-type estimates linking spectral properties with graph structure.

Findings

Criteria for quantum graphs to have positive or discrete spectra

Connections established between quantum and combinatorial isoperimetric constants

Applications to trees, antitrees, and Cayley graphs of groups

Abstract

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.