Transcription of Spectral and Algebraic Graph Theory
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Spectral and Algebraic Graph Theory
Spectral and Algebraic Graph TheoryIncomplete Draft, dated December 4, 2019 Current version available A. SpielmanYale UniversityCopyrightc 2019 by Daniel A. Spielman. All rights ListPrefacevContentsviNotationxxiiI Introduction and Background11 Introduction22 Eigenvalues and Optimization: The Courant-Fischer Theorem213 The Laplacian and Graph Drawing274 Adjacency matrices, Eigenvalue Interlacing, and the Perron-Frobenius Theorem 325 Comparing Graphs39II The Zoo of Graphs466 Fundamental Graphs477 Cayley Graphs558 Eigenvalues of Random Graphs639 Strongly Regular Graphs73iCHAPTER LISTiiIII Physical Metaphors8210 Random Walks on Graphs8311 Walks, Springs, and Resistor Networks9312 Effective Resistance and Schur Complements10113 Random Spanning Trees11014 Approximating Effective Resistances11715 Tutte s Theorem: How to draw a graph12216 The Lov`asz - Simonovits Approach to Random Walks13017 Monotonicity and its Failures13518 Dynamic and Nonlinear Networks143IV Spectra and Graph Structure15119 Independent Sets and Coloring15220 Graph Partitioning15921 Cheeger s Inequality16422 Local Graph Clustering16923 Spectral Partitioning in a Stochastic Block Model17724 Nodal Domains18425 The Second Eigenvalue of Planar Graphs19226 Planar Graphs 2, the Colin de Verdi`ere Number1
Chapter List Preface v Contents vi Notation xxii I Introduction and Background1 1 Introduction 2 2 Eigenvalues and Optimization: The Courant-Fischer Theorem21 3 The Laplacian and Graph Drawing27 4 Adjacency matrices, Eigenvalue Interlacing, and the Perron-Frobenius Theorem32 5 Comparing Graphs39 II The Zoo of Graphs46 6 Fundamental Graphs47 7 ...
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