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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.

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 Cayley Graphs 55 8 Eigenvalues of Random Graphs63

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