Transcription of Eigenvalues and the Laplacian of a graph - UCSD Mathematics
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Eigenvalues and the Laplacian of a graph - UCSD Mathematics
CHAPTER 1 Eigenvalues and the Laplacian of a IntroductionSpectral graph theory has a long history. In the early days, matrix theory andlinear algebra were used to analyze adjacency matrices of graphs. Algebraic meth-ods have proven to be especially effective in treating graphs which are regular andsymmetric. Sometimes, certain Eigenvalues have been referred to as the algebraicconnectivity of a graph [127]. There is a large literature on algebraic aspects ofspectral 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 oftenhad a geometric flavor.
underlying mathematics of spectral graph theory through all its connections to the pure and applied, the continuous and discrete, can be viewed as a single uni ed subject. It is this aspect that we intend to cover in this book. 1.2. The Laplacian and eigenvalues Before we start to de ne eigenvalues, some explanations are in order. The
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