Spectral analysis of metric graphs and related spaces. (English) Zbl 1219.05089
Arzhantseva, Goulnara (ed.) et al., Limits of graphs in group theory and computer science. Lausanne: EPFL Press/distrib. by CRC Press (ISBN 978-1-4398-0400-1/hbk; 978-2-940222-33-9). Fundamental Sciences. Mathematics, 109-140 (2009).
Summary: 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.
For the entire collection see [Zbl 1166.05001].
For the entire collection see [Zbl 1166.05001].
MSC:
05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
05C10 | Planar graphs; geometric and topological aspects of graph theory |
05C12 | Distance in graphs |
05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |