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References:
1. Cornell University Lecture Note: https://people.orie.cornell.edu/dpw/orie6334/Fall2016/lecture7.pdf
2. Spectral Graph Theory Lecture by Steve Butler: https://www.stevebutler.org/spectral2023
3. Spectral Graph Theory Lecture by Radu Horaud: https://csustan.csustan.edu/~tom/Clustering/GraphLaplacian-tutorial.pdf
4. Tutorial by Daniel A. Spielman: https://www.cs.cmu.edu/afs/cs/user/glmiller/public/Scientific-Computing/F-11/RelatedWork/Spielman/SpectTut.pdf
5. This Stack Exchange question: https://stats.stackexchange.com/questions/459640/why-eigenvectors-reveal-the-groups-in-spectral-clustering
6. And this Quora question: https://www.quora.com/Whats-the-intuition-behind-a-Laplacian-matrix-Im-not-so-much-interested-in-mathematical-details-or-technical-applications-Im-trying-to-grasp-what-a-laplacian-matrix-actually-represents-and-what-aspects-of-a-graph-it-makes-accessible
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Timestamp:
0:00 Introduction
0:30 Outline
00:57 Review of Graph Definition and Degree Matrix
03:34 Adjacency Matrix Review
05:03 Review of Necessary Linear Algebra
09:09 Introduction of The Laplacian Matrix
15:36 Why is L called the Laplace Matrix
18:14 Eigenvalue 0 and Its Eigenvector
20:27 Fiedler Eigenvalue and Eigenvector
23:56 Sponsorship Message
25:02 Spectral Embedding
25:38 Spectral Embedding Application: Spectral Clustering
27:51 Outro
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Big thanks to professor Fan Chung Graham and professor Robert Ellis for the email conversations.