Spectral analysis of metric graphs and related spaces
arXiv:0712.1507
Abstract
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger's inequality and trace formulas for metric and discrete (generalised) Laplacians.
arXiv comment24 pages, extended version of a lecture held at the EPF Lausanne, some references added and typos corrected
References in corpus (7)
- Can One Hear the Shape of a Graph?
- Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
- Spectral convergence of non-compact quasi-one-dimensional spaces
- Spectra of Schroedinger operators on equilateral quantum graphs
- Equilateral quantum graphs and boundary triples
- Spectral Convergence of the Discrete Laplacian on Models of a Metrized Graph
- First order approach and index theorems for discrete and metric graphs