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Spectral and Algebraic Graph Theory - Yale University

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

One must introduce necessary linear algebra and show some interesting interpretations of graph eigenvalues. One must derive the eigenvalues of some example graphs to ground the theory. I nd that one has to do all these at once. For this reason my rst few lectures jump between developing theory and examining particular graphs.

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