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This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own.
Perhaps surprisingly, eigenvalues and eigenvectors turn out to be intimately connected with the structure of a graph. In terms of applications, they have proved to be useful for isomorphism testing and embedding graphs in the plane, for graph partitioning and clustering, as topological descriptors for networks and molecules, in the geometric description of data sets in Data Science, and in the design of efficient networks, just to mention a few. In a purely mathematical perspective, the study of graph spectra has led to a myriad of open problems, ranging from the construction of graphs with a given set of eigenvalues to extremal problems that ask for a characterization of graphs that maximize or minimize some spectral parameter.
Of course, computing these eigenvalues and eigenvectors is a necessary step in any such application. Since eigenvalues are the roots of a polynomial, in general we cannot expect to find simple expressions for these roots. However, there are numerical algorithms that allow us to approximate them with any desired precision in polynomial time.
In this book, we survey the evolution of eigenvalue location algorithms in an organized and unified way, starting with algorithms for trees and other well-known graph classes, such as cographs, and showing how they motivated more recent algorithms that may be applied to arbitrary graphs, but whose efficiency depends on the existence of a graph decomposition of low complexity. While they are vastly deeper than the simple tree algorithm, we wish to convince the readers that they are similar in spirit