AMS, 1997. — 215 p.This monograph is an intertwined tale of eigenvalues and their use in unlocking a thousand secrets about graphs. The stories will be told - how the spectrum reveals fundamental properties of a graph, how spectral graph theory links the discrete universe to the continuous one through geometric, analytic and algebraic techniques, and how, through eigenvalues, theory and applications in communications and computer science come together in sy mbiotic harmony. Since spectral graph theory has been evolving very rapidly, the above goals can only be partially fulfilled here. For example, more advanced material on the heat kernel will be treated in a later publication.Eigenvalues and the Laplacian of a graph. Isoperimetric problems. Diameters and eigenvalues. Paths, flows, and routing. Eigenvalues and quasi-randomness. Expanders and explicit constructions. Eigenvalues of symmetrical graphs. Eigenvalues of subgraphs with boundary conditions. Harnack inequalities. Heat kernels. Sobolev inequalities. Advanced techniques for random walks on graphs.
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