Spectral Graph Theory-Summer 2015

 Spectral graph theory  mostly deals with spectrum of the Laplacian of a graph. While graph Laplacians have many similarities with Laplacians of Riemannian manifolds, the study of them requires less technical know how. In particular, while mathematical sophistication will be essential, the only prerequisite for this course is a good understanding of linear algebra and multivariable calculus. No prior familiarity with graph theory will be assumed. The advantage of working with graphs is that many ideas of spectral geometry and analytic numbers theory can be studied in a very concrete manner. With the ever increasing role played by large networks and big data in every day life, ideas developed in spectral graph theory are gaining importance in many fields now.

Note: this is not a graph theory course!

 Time permitting, we shall try to cover some (hopefully all!) of the following topics:

Laplacian of a graph and its eigenvalues,

Isospectral but not isometric graphs,

Cheeger inequlaity,

Random walks on graphs, Markov chains, Approach to equilibrium, Ergodicity,

Dirichlet boundary value problems for graphs,

Phase transition in planar Ising model,

Matrix-tree theorem,

Heat kernel techniques,

Counting periodic orbits, Ihara zeta function of a graph, and its Riemann hypothesis. 

Marking Policy:  70% assignments,  30% project. Projects will be chosen in consultation with each student, and usually reflect student’s area of specialization or interests. I expect students to prepare a short essay on their project and present it in class in a 50 minutes lecture towards the end of the term. Students should make sure to see me no later than 3 weeks after the start of the class regarding their projects.


Linear algebra, mathematical maturity and sophistication.

Topics per day:

You can find pdf file of the topics Discussed in each class here.


Student Presentations


Here are the topics and notes: