Spectral Estimates for Infinite Quantum Graphs

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.

38 pages; 2 figures

References in corpus (1)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators

Cited by in corpus (5)

• Self-adjoint and Markovian extensions of infinite quantum graphs
• A note on the Gaffney Laplacian on infinite metric graphs
• Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap
• Some spectral comparison results on infinite quantum graphs
• The $k$-adjacency operators and adjacency Jacobi matrix on distance-regular graphs