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Unformatted text preview: CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. Introduction Spectral graph theory has a long history. In the early days, matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Algebraic meth- ods have proven to be especially effective in treating graphs which are regular and symmetric. Sometimes, certain eigenvalues have been referred to as the algebraic connectivity of a graph [ 127 ]. There is a large literature on algebraic aspects of spectral graph theory, well documented in several surveys and books, such as Biggs [ 26 ], Cvetkovi c, Doob and Sachs [ 93 ] (also see [ 94 ]) and Seidel [ 228 ]. In the past ten years, many developments in spectral graph theory have often had a geometric flavor. For example, the explicit constructions of expander graphs, due to Lubotzky-Phillips-Sarnak [ 197 ] and Margulis [ 199 ], are based on eigenvalues and isoperimetric properties of graphs. The discrete analogue of the Cheeger in- equality has been heavily utilized in the study of random walks and rapidly mixing Markov chains [ 228 ]. New spectral techniques have emerged and they are powerful and well-suited for dealing with general graphs. In a way, spectral graph theory has entered a new era. Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction. We will see that eigenvalues are closely related to almost all major invariants of a graph, linking one extremal property to another. There is no question that eigenvalues play a central role in our fundamental understanding of graphs. The study of graph eigenvalues realizes increasingly rich connections with many other areas of mathematics. A particularly important development is the interac- tion between spectral graph theory and differential geometry. There is an interest- ing analogy between spectral Riemannian geometry and spectral graph theory. The concepts and methods of spectral geometry bring useful tools and crucial insights to the study of graph eigenvalues, which in turn lead to new directions and results in spectral geometry. Algebraic spectral methods are also very useful, especially for extremal examples and constructions. In this book, we take a broad approach with emphasis on the geometric aspects of graph eigenvalues, while including the algebraic aspects as well. The reader is not required to have special background in geometry, since this book is almost entirely graph-theoretic....
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