Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 621 Wiki page.

1. Relations between properties of an undirected graph and the eigenvalues of its adjacency matrix.

2. Spectra of several simple classes of graphs: complete graphs, paths, cycles, stars, etc.

3. Theory of nonnegative matrices to spectral graph theory.

4. Bipartite graph in terms of its graph spectrum.

5. Graph parameters such as the clique number and chromatic number.

6. Basic properties of the Laplacian matrix of a graph.

7. Matrix tree theorem.

8.  Laplacian matrix.