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.