Spectral Graph Theory: Foundations, Algorithms, and Applications
This course provides a comprehensive introduction to graph theory with a focus on spectral methods, exploring the deep connections between graphs and linear algebra. Students will learn the fundamental concepts of graph theory, including adjacency and Laplacian matrices, eigenvalues, and eigenvectors, and their applications in analyzing graph structure and behavior. The course will cover key theoretical results such as Cheeger's inequality, graph expansion, and random walks, as well as practical applications in network analysis, clustering, and machine learning. Through a combination of lectures, problem sets, and projects, students will gain both theoretical understanding and hands-on experience with spectral graph theory. The course is designed to cater to both undergraduate and graduate students, with an emphasis on intuition, proofs, and real-world applications.
Lecturer
Date
24th February ~ 26th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 16:55 | A7-303 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
To succeed in this course, students should have a solid foundation in the following areas: Linear Algebra: Familiarity with matrices, eigenvectors, eigenvalues, and basic matrix operations. Understanding of diagonalization and spectral decomposition. Discrete Mathematics:Basic knowledge of combinatorics, graph theory fundamentals (e.g., paths, cycles, trees), and proof techniques (induction, contradiction). Calculus: Comfort with limits, derivatives, and integrals (useful for understanding probabilistic methods and continuous analogs.
Syllabus
Week 1: Introduction to Graph Theory & Overview of Spectral Graph Theory
Basic graph concepts: vertices, edges, degree, adjacency, Laplacian matrices
Introduction to spectral graph theory and its applications
Week 2: Eigenvalues of the Adjacency Matrix
Definition and properties of the adjacency matrix
Relationship between eigenvalues and graph structure
Week 3: The Laplacian Matrix
Definition and properties of the Laplacian matrix
Spectral properties of the Laplacian matrix and its applications to graph connectivity
Week 4: Random Walks and Spectral Analysis of Graphs
Basic concepts of random walks on graphs
Connection between random walks and spectral graph theory
Week 5: Expander Graphs and Spectral Gaps
Definition and applications of expander graphs
Spectral gaps and their role in graph expansion properties
Week 6: Cheeger’s Inequality and Graph Connectivity
Derivation and application of Cheeger’s inequality
Relationship between spectral gaps and graph connectivity
Week 7: Spectral Graph Partitioning
Introduction to spectral partitioning methods
Graph partitioning using the Laplacian matrix
Week 8: Markov Chains and Spectral Graph Theory
Relationship between Markov chains and spectral graph theory
Spectral properties of Markov chains and mixing times
Week 9: Spectral Properties of Special Graphs
Spectral properties of trees, bipartite graphs, and regular graphs
Spectral analysis of specific graph classes
Week 10: Advanced Applications of Spectral Graph Theory
Applications of spectral graph theory in machine learning
Spectral analysis of large-scale graphs
Week 11: Practical Applications of Spectral Graph Theory
Applications in network science, image processing, and computational biology
Week 12: Review & Future Directions
Recap of key concepts and methods covered in the course
Discussion of the future directions and research areas in spectral graph theory
Basic graph concepts: vertices, edges, degree, adjacency, Laplacian matrices
Introduction to spectral graph theory and its applications
Week 2: Eigenvalues of the Adjacency Matrix
Definition and properties of the adjacency matrix
Relationship between eigenvalues and graph structure
Week 3: The Laplacian Matrix
Definition and properties of the Laplacian matrix
Spectral properties of the Laplacian matrix and its applications to graph connectivity
Week 4: Random Walks and Spectral Analysis of Graphs
Basic concepts of random walks on graphs
Connection between random walks and spectral graph theory
Week 5: Expander Graphs and Spectral Gaps
Definition and applications of expander graphs
Spectral gaps and their role in graph expansion properties
Week 6: Cheeger’s Inequality and Graph Connectivity
Derivation and application of Cheeger’s inequality
Relationship between spectral gaps and graph connectivity
Week 7: Spectral Graph Partitioning
Introduction to spectral partitioning methods
Graph partitioning using the Laplacian matrix
Week 8: Markov Chains and Spectral Graph Theory
Relationship between Markov chains and spectral graph theory
Spectral properties of Markov chains and mixing times
Week 9: Spectral Properties of Special Graphs
Spectral properties of trees, bipartite graphs, and regular graphs
Spectral analysis of specific graph classes
Week 10: Advanced Applications of Spectral Graph Theory
Applications of spectral graph theory in machine learning
Spectral analysis of large-scale graphs
Week 11: Practical Applications of Spectral Graph Theory
Applications in network science, image processing, and computational biology
Week 12: Review & Future Directions
Recap of key concepts and methods covered in the course
Discussion of the future directions and research areas in spectral graph theory
Reference
Primary Textbook:
Chung, F. R. K. (1997). Spectral Graph Theory. American Mathematical Society.
A classic and accessible introduction to spectral graph theory, suitable for both undergraduates and graduates.
Supplementary Textbooks:
Diestel, R. (2017). Graph Theory (5th ed.). Springer.
A comprehensive reference for foundational graph theory concepts.
Godsil, C., & Royle, G. (2001). Algebraic Graph Theory. Springer.
A more advanced text focusing on algebraic methods in graph theory.
Spielman, D. A. (2009). Spectral Graph Theory and its Applications. Lecture Notes, Yale University.
A concise and practical set of lecture notes with a focus on algorithms and applications.
Additional Resources:
Newman, M. (2018). Networks (2nd ed.). Oxford University Press.
A great resource for applications of graph theory in network science.
Leskovec, J., Rajaraman, A., & Ullman, J. D. (2020). Mining of Massive Datasets (3rd ed.). Cambridge University Press.
Covers spectral clustering and graph-based algorithms in data mining.
Von Luxburg, U. (2007). A Tutorial on Spectral Clustering. Statistics and Computing, 17(4), 395–416.
A highly cited paper on spectral clustering techniques.
Chung, F. R. K. (1997). Spectral Graph Theory. American Mathematical Society.
A classic and accessible introduction to spectral graph theory, suitable for both undergraduates and graduates.
Supplementary Textbooks:
Diestel, R. (2017). Graph Theory (5th ed.). Springer.
A comprehensive reference for foundational graph theory concepts.
Godsil, C., & Royle, G. (2001). Algebraic Graph Theory. Springer.
A more advanced text focusing on algebraic methods in graph theory.
Spielman, D. A. (2009). Spectral Graph Theory and its Applications. Lecture Notes, Yale University.
A concise and practical set of lecture notes with a focus on algorithms and applications.
Additional Resources:
Newman, M. (2018). Networks (2nd ed.). Oxford University Press.
A great resource for applications of graph theory in network science.
Leskovec, J., Rajaraman, A., & Ullman, J. D. (2020). Mining of Massive Datasets (3rd ed.). Cambridge University Press.
Covers spectral clustering and graph-based algorithms in data mining.
Von Luxburg, U. (2007). A Tutorial on Spectral Clustering. Statistics and Computing, 17(4), 395–416.
A highly cited paper on spectral clustering techniques.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
No
Notes Public
Yes
Language
Chinese
, English