### Lecture Notes and Readings

The book, *Spectral and Algebraic Graph Theory*, by Daniel Spielman, from which many of the readings are taken, can be downloaded here. The chapter listings refer to the December 4, 2019 version of the book draft.

All lectures in one pdf: here

- Lecture 01

Introduction to Spectral Graph Theory

Date: January 19, 2021

Reading: None - Lecture 02

Notebooks: path graph frequency, Minnesota graph embedding

Introduction to Spectral Graph Theory (Part 2)

Date: January 21, 2021

Reading: Chapter 1 of*Spectral and Algebraic Graph Theory* - Lecture 03

Eigenvalues, Optimization, and Connectivity

Date: January 26, 2021

Reading: Chapter 2 of*Spectral and Algebraic Graph Theory* - Lecture 04

The Complete Graph and Drawing Graphs

Date: January 28, 2021

Reading: Chapters 3 and 6.1 of*Spectral and Algebraic Graph Theory* - Lecture 05

Product Graphs and Star Graphs

Date: February 2, 2021

Reading: Chapters 6.2 and 6.3 of*Spectral and Algebraic Graph Theory* - Lecture 06

Notebooks: path graph test vector

Test Vectors and Comparing Graphs

Date: February 4, 2021

Reading: Chapters 5 and 6.4 of*Spectral and Algebraic Graph Theory* - Lecture 07

The Cycle Graph and the Path Graph

Date: February 9, 2021

Reading: Chapters 6.5 and 6.6 of*Spectral and Algebraic Graph Theory* - Lecture 08

The Adjacency Matrix and Eigenvalue Interlacing

Date: February 11, 2021

Reading: Chapters 4.1-4.4 of*Spectral and Algebraic Graph Theory* - Lecture 09

Perron-Frobenius Theory

Date: February 16, 2021

Reading: Chapter 4.5 of*Spectral and Algebraic Graph Theory* - Lecture 10

The Weighted Path Graph

Date: February 18, 2021

Reading: Chapters 24.1-24.3 of*Spectral and Algebraic Graph Theory* - Lecture 11

Fiedler’s Nodal Domain Theorem

Date: February 23, 2021

Reading: Chapters 24.4 and 24.5 of*Spectral and Algebraic Graph Theory* - Lecture 12

Graph Signal Processing, Part I

Date: February 25, 2021

Reading: Read from the beginning of The Emerging Field of Signal Processing on Graphs through the section*Other Graph Matrices*(inclusive). - Lecture 13

Graph Signal Processing, Part II

Date: March 4, 2021

Reading: In the paper The Emerging Field of Signal Processing on Graphs, read from the section*Generalized Operators for Signals on Graphs*through the section*Translation*(inclusive of both sections). - Lecture 14

Graph Signal Processing, Part III

Date: March 9, 2021

Reading: Finish reading the paper The Emerging Field of Signal Processing on Graphs. - Lecture 15

Introduction to Graph Partitioning and Clustering

Date: March 11, 2021

Reading: Chapter 20.1 of*Spectral and Algebraic Graph Theory* - Lecture 16

Graph Conductance

Date: March 16, 2021

Reading: Chapters 20.2-20.4 of*Spectral and Algebraic Graph Theory* - Lecture 17

Preliminaries to Cheeger’s Inequality

Date: March 18, 2021

Reading: None - Lecture 18

Centered Vectors

Date: March 23, 2021

Reading: None - Lecture 19

Cheeger’s Inequality and Spectral Clustering

Date: March 25, 2021

Reading: Chapter 21 of*Spectral and Algebraic Graph Theory*Optional reading: Chapter 23 of*Spectral and Algebraic Graph Theory*, which discusses the two-cluster stochastic block model and gives theoretical guarantees for when one can recover the two clusters using the second eigenvector of the adjacency matrix. - Lecture 20

Random Walks on Graphs, Part I

Date: March 30, 2021

Reading: Chapters 10.1 and 10.2 of*Spectral and Algebraic Graph Theory* - Lecture 21

Random Walks on Graphs, Part II

Date: April 1, 2021

Reading: Chapters 10.3-10.8 of*Spectral and Algebraic Graph Theory*

Optional reading: Chapter 22 of*Spectral and Algebraic Graph Theory*, which discusses local clustering algorithms. The algorithm presented in chapter 22 synthesizes what we learned about random walks on graphs with Cheeger’s inequality! - Lecture 22

Expander Graphs, Part I

Date: April 6, 2021

Reading: Chapters 27.1-27.3 of*Spectral and Algebraic Graph Theory* - Lecture 23

Expander Graphs, Part II

Date: April 8, 2021

Reading: Chapters 27.4-27.6 of*Spectral and Algebraic Graph Theory*

Optional reading: If you want to learn more about expander graphs, check out these readings from*Spectral and Algebraic Graph Theory:*- Chapter 30: Constructions of expander graphs
- Chapters 28-29: Application of expander graphs to constructing good error-correcting codes
- Chapter 31: Application of expander graphs to pseudo-random generators

- Lecture 24

Graph Sparsification via Random Sampling, Part I

Date: April 13, 2021

Reading: Chapters 32.1-32.3 of*Spectral and Algebraic Graph Theory* - Lecture 25

Graph Sparsification via Random Sampling, Part II

Date: April 15, 2021

Reading: Chapters 32.4-32.7 of*Spectral and Algebraic Graph Theory*

Optional reading: Chapter 33 of*Spectral and Algebraic Graph Theory*, which describes how to come up with an epsilon-approximation of a graph with only O(epsilon^{-2} n) edges, thus dropping the log(n) scaling from our random sampling result. - Lecture 26

Graph Convolution Networks

Date: April 20, 2021

Reading: References [6, 7, 8] in the lecture notes.