Spectral and Algebraic Graph Theory (Draft)

399 Pages·2019·2.782 MB·English

by  Daniel A. Spielman

Spectral and Algebraic Graph Theory
Incomplete Draft, dated December 4, 2019
Current version available at http://cs-www.cs.yale.edu/homes/spielman/sagt.
Daniel A. Spielman
Yale University
Copyright c 2019 by Daniel A. Spielman. All rights reserved.
(cid:13)
Chapter List
Preface v
Contents vi
Notation xxii
I Introduction and Background 1
1 Introduction 2
2 Eigenvalues and Optimization: The Courant-Fischer Theorem 21
3 The Laplacian and Graph Drawing 27
4 Adjacency matrices, Eigenvalue Interlacing, and the Perron-Frobenius Theorem 32
5 Comparing Graphs 39
II The Zoo of Graphs 46
6 Fundamental Graphs 47
7 Cayley Graphs 55
8 Eigenvalues of Random Graphs 63
9 Strongly Regular Graphs 73
i
CHAPTER LIST ii
III Physical Metaphors 82
10 Random Walks on Graphs 83
11 Walks, Springs, and Resistor Networks 93
12 Effective Resistance and Schur Complements 101
13 Random Spanning Trees 110
14 Approximating Effective Resistances 117
15 Tutte’s Theorem: How to draw a graph 122
16 The Lov`asz - Simonovits Approach to Random Walks 130
17 Monotonicity and its Failures 135
18 Dynamic and Nonlinear Networks 143
IV Spectra and Graph Structure 151
19 Independent Sets and Coloring 152
20 Graph Partitioning 159
21 Cheeger’s Inequality 164
22 Local Graph Clustering 169
23 Spectral Partitioning in a Stochastic Block Model 177
24 Nodal Domains 184
25 The Second Eigenvalue of Planar Graphs 192
26 Planar Graphs 2, the Colin de Verdi`ere Number 199
CHAPTER LIST iii
V Expander Graphs 206
27 Properties of Expander Graphs 207
28 A brief introduction to Coding Theory 216
29 Expander Codes 224
30 A simple construction of expander graphs 231
31 PSRGs via Random Walks on Graphs 239
VI Algorithms 245
32 Sparsification by Random Sampling 246
33 Linear Sized Sparsifiers 252
34 Iterative solvers for linear equations 260
35 The Conjugate Gradient and Diameter 268
36 Preconditioning Laplacians 275
37 Augmented Spanning Tree Preconditioners 283
38 Fast Laplacian Solvers by Sparsification 288
39 Testing Isomorphism of Graphs with Distinct Eigenvalues 295
40 Testing Isomorphism of Strongly Regular Graphs 305
VII Interlacing Families 313
41 Bipartite Ramanujan Graphs 314
42 Expected Characteristic Polynomials 329
CHAPTER LIST iv
43 Quadrature for the Finite Free Convolution 336
44 Ramanujan Graphs of Every Size 344
45 Matching Polynomials of Graphs 352
46 Bipartite Ramanujan Graphs of Every Degree 358
Bibliography 363
Index 375
Preface
Please note that this is a rapidly evolving draft.
This book is about how combinatorial properties of graphs are related to algebraic properties of
associated matrices, as well as applications of those connections. One’s initial excitement over this
material usually stems from its counter-intuitive nature. I hope to convey this initial amazement,
but then make the connections seem intuitive. After gaining intuition, I hope the reader will
appreciate the material for its beauty.
This book is mostly based on lecture notes from the “Spectral Graph Theory” course that I have
taught at Yale, with notes from “Graphs and Networks” and “Spectral Graph Theory and its
Applications” mixed in. I love the material in these courses, and find that I can never teach
everything I want to cover within one semester. This is why I am have written this book. As this
book is based on lecture notes, it does not contain the tightest or most recent results. Rather, my
goal is to introduce the main ideas and to provide intuition.
There are three tasks that one must accomplish in the beginning of a course on Spectral Graph
Theory:
One must convey how the coordinates of eigenvectors correspond to vertices in a graph.
•
This is obvious to those who understand it, but it can take a while for students to grasp.
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 find that one has to do all these at once. For this reason my first few lectures jump between
developing theory and examining particular graphs. For this book I have decided to organize the
material differently, mostly separating examinations of particular graphs from the development of
the theory. To help the reader reconstruct the flow of my courses, I give three orders that I have
used for the material:
put orders here
There are many terrific books on Spectral Graph Theory. The four that influenced me the most
are
“Algebraic Graph Theory” by Norman Biggs,
v
PREFACE vi
“Spectral Graph Theory” by Fan Chung,
“Algebraic Combinatorics” by Chris Godsil, and
“Algebraic Graph Theory” by Chris Godsil and Gordon Royle.
Other books that I find very helpful and that contain related material include
“Modern Graph Theory” by Bela Bollobas,
“Probability on Trees and Networks” by Russell Llyons and Yuval Peres,
“Spectra of Graphs” by Dragos Cvetkovic, Michael Doob, and Horst Sachs, and
“Eigenspaces of Graphs” By Dragos Cvetkovic, Peter Rowlinson, and Slobodan Simic”
“Non-negative Matrices and Markov Chains” by Eugene Seneta
“Nonnegative Matrices and Applications” by R. B. Bapat and T. E. S. Raghavan
“Numerical Linear Algebra” by Lloyd N. Trefethen and David Bau, III
“Applied Numerical Linear Algebra” by James W. Demmel
For those needing an introduction to linear algebra, a perspective that is compatible with this
book is contained in Gil Strang’s “Introduction to Linear Algebra.” For more advanced topics in
linear algebra, I recommend “Matrix Analysis” by Roger Horn and Charles Johnson, as well as
their “Topics in Matrix Analysis” For treatments of physical systems related to graphs, the topic
of Part III, I recommend Gil Strang’s “Introduction to Applied Mathematics”, Sydney H. Gould’s
“Variational Methods for Eigenvalue Problems”, and “Markov Chains and Mixing Times” by
Levin, Peres and Wilmer.
I include some example in these notes. All of these have been generated inside Jupyter notebooks
using the Julia language. Some of them require use of the package Laplacians.jl. A simple search
will produce good instructions for installing Julia and packages for it. The notebooks used in this
book may be found at http://cs-www.cs.yale.edu/homes/spielman/sagt.
Contents
Preface v
Contents vi
Notation xxii
I Introduction and Background 1
1 Introduction 2
1.1 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Matrices for Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 A spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 An operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 A quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 Spectral Graph Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.2 Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.3 Platonic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.4 The Fiedler Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.5 Bounding Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
vii
CONTENTS viii
1.5.6 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.7 Random Walks on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.8 Expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.9 Approximations of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.10 Solving equations in and computing eigenvalues of Laplacians . . . . . . . . . 18
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Eigenvalues and Optimization: The Courant-Fischer Theorem 21
2.1 The First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Proof of the Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 The Laplacian and Graph Drawing 27
3.1 The Laplacian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Drawing with Laplacian Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Adjacency matrices, Eigenvalue Interlacing, and the Perron-Frobenius Theorem 32
4.1 The Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 The Largest Eigenvalue, µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
4.3 Eigenvalue Interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Wilf’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Perron-Frobenius Theory for symmetric matrices . . . . . . . . . . . . . . . . . . . . 36
5 Comparing Graphs 39
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 The Loewner order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Approximations of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4 The Path Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4.1 Bounding λ of a Path Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2
5.5 The Complete Binary Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
CONTENTS ix
5.6 The weighted path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
II The Zoo of Graphs 46
6 Fundamental Graphs 47
6.1 The complete graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 The star graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3 Products of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3.1 The Hypercube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.4 Bounds on λ by test vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2
6.5 The Ring Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.6 The Path Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7 Cayley Graphs 55
7.1 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Paley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Eigenvalues of the Paley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4 Generalizing Hypercubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5 A random set of generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.7 Non-Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.8 Eigenvectors of Cayley Graphs of Abelian Groups . . . . . . . . . . . . . . . . . . . . 62
8 Eigenvalues of Random Graphs 63
8.1 Transformation and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.2 The extreme eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.3 Expectation of the trace of a power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.4 The number of walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71