This course aims to introduce Algebraic Graph Theory and provide students with a comprehensive understanding of the subject. The course focuses on analyzing the algebraic structure associated with a graph and covers a range of topics, including the adjacency matrix of a graph and its eigenvalues, Bose-Mesner algebra, dual Bose-Mesner algebra, Terwilliger algebra and its modules, distance regular graphs, the Q-polynomial property, and the Askey-Wilson duality. Additionally, we will explore connections with Lie theory, representation theory, quantum groups, and orthogonal polynomials.

Undergraduate Linear Algebra

- Attendance: 10%
- Homework: 60%
- Final Presentation: 30%

- E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, 1984.
- A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
- N. Biggs, Algebraic Graph Theory, Cambridge Mathematical Library, 2nd ed, 1993.

Week 1: Introduction to graphs and their spectra
Week 2: The adjacency algebra and dual adjacency algebra, the Terwilliger algebra T
Week 3: The irreducible T-modules, The hypercube and its spectrum
Week 4: The tridiagonal pairs, the Perron-Fronenius theorem
Week 5: Distance-regular graphs
Week 6: The intersection numbers, the Norman Biggs multiplicity formula
Week 7: The representations of a distance-regular graph, the Krawtchouk polynomials
Week 8: The Q-polynomial property and Askey-Wilson duality
Week 9: The tridiagonal relations (1)
Week 10: The tridiagonal relations (2)
Week 11: Q-polynomial distance-regular graphs
Week 12: The primary T-module and the Askey-Wilson relations
Week 13: Characterization of the Q-polynomial property
Week 14: Distance-regular graphs with classical parameters
Week 15: Student final presentation (1)
Week 16: Student final presentation (2)

The class primarily consists of self-contained lectures, with the final two weeks of the semester allocated for student presentations.
During this time, each student is expected to give one lecture on a topic of their choice related to the course.
As the time approaches, I will suggest topics and organize the speaking schedule.

We aim to master the materials covered in the lecture. To achieve this, I will assign the following homework every week: at the end of each week's lectures, students must summarize the content covered during the week and submit their notes. Specifically, students are expected to handwrite and submit detailed portions of the proofs covered in class. Additionally, to master the course material, it is recommended to find and work through many examples. Moreover, if I have proven something to be true for all n, try to verify it by directly calculating specific smaller values of n. Furthermore, if possible, try to prove the lemmas/theorems covered in the lecture using your own methods. Whether your proof matches mine is not important. If you are currently conducting your research, explore whether the materials in the lectures are connected to your research. Make conjectures and try to prove them.