Spectra of graphs. (Spectres de graphes.) (French) Zbl 0913.05071

Colin de Verdière, Yves

Cours Spécialisés. 4. Paris: Société Mathématique de France. vi, 114 p. (1998).

After a book by the reviewer, M. Doob and H. Sachs [Spectra of graphs. Theory and application, Deutscher Verlag der Wissenschaften and Academic Press (1980; Zbl 0458.05042)] and a book by F. R. K. Chung [Spectral graph theory, American Mathematical Society (1997; Zbl 0867.05046)] the book under review is the third book on graph spectra. As the author says in the preface, the intersection with the previous books is almost empty. In fact, contrary to the first book, the intention was not to give a complete treatment of the subject. The author, like the author of the second book, is led by his own points of interest. Besides a preface, an introduction, a bibliography with 76 references and a condensed subject index, the book contains the following chapters: 1. Definitions and examples, 2. Spectra, 3. Spectral gap of graphs and their expanding properties, 4. Singular limits and gamma convergence, 5. Eigenvalue multiplicities and related invariants, 6. Discrete and continual, 7. Electrical networks. The aim of the book is to develop for finite graphs some analogues of the spectral theory of Schrödinger operators on compact manifolds. The book includes canonical Laplacians on graphs, Cheeger’s inequalities, Markov processes, finite element methods and many other interesting topics.

Reviewer: D.Cvetković (Beograd)

MSC:

05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)
05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
05C15	Coloring of graphs and hypergraphs
35J10	Schrödinger operator, Schrödinger equation
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

graph spectrum; graph embedding; singular perturbation of eigenvalues; tunneling; finite elements; Schrödinger equation

Citations:

Zbl 0458.05042; Zbl 0867.05046