- Eigenvalues and eigenvectors of common graphs: paths, trees, cliques, grids, products.
- Random walks on graphs.
- Eigenvalues and the diameter of a graph.

- Cheeger's inequality.
- Hoffman's bound on the size of an independent set in a graph.
- Approximations of graphs and sparsification.
- Low-stretch spanning tress and generalized eigenvalues of graphs.

- Expander graphs, with applications in coding theory and derandomization.
- Isoperimetry, with applications in probability.
- Eigenvalues of random graphs.
- Finding cliques in and partitioning semi-random graphs.
- Testing isomorphism of graphs of bounded eigenvalue multiplicity.

- Constructions of higly symmetric graphs: strongly regular graphs and expanders.
- Eigenvalues of planar graphs.
- The Colin de Verdière number of a graph.
- Spectral theory for directed graphs.

This course will assume familiarity with graphs and linear algebra. While this background knowledge is elementary, the course will move at a fast pace. This course may be suitable for advanced undergraduates. Undergraduate students who are interested in taking the course are advised to consult with the instructor before registering.