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A survey on spectra of infinite graphs. (English) Zbl 0645.05048
An overview on spectra of locally finite infinite graphs and related topics is presented. The adjacency matrix and the transition matrix of the simple random walk on a graph, and the corresponding difference Laplacian matrices and their spectral properties are considered. There are eight sections: Introduction, Linear operators associated with a graph, Basic results, Spectral radius, walk generating functions and spectral measures, Growth and isoperimetric number, Positive eigenfunctions, Graphs of groups, distance regular graphs and trees, Some remarks on applications in chemistry and physics. This paper extends the disposition of related topics in the recent monograph [D. M. Cvetković, M. Doob, I. Gutman and A. Torgašev, Recent results in the theory of graph spectra (1988; Zbl 0634.05054)] where a completely non-combinatorial approach to the spectrum of infinite graphs is presented.
Reviewer: B.Mohar

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15B48 Positive matrices and their generalizations; cones of matrices
47A10 Spectrum, resolvent
60G50 Sums of independent random variables; random walks
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