Minicourse 1: Algebraic Graph Theory: an introduction to eigenvalues and the ratio bound with a focus on examples
Abstract: This course will be an introduction to Algebraic Graph Theory.
We will discuss automorphism groups of graphs, cliques and cocliques. We will define eigenvalues of graphs and see several examples, including Cayley graphs. We will show different techniques to find eigenvalues, including equitable partitions and quotient graphs. We will build up to the proof of the Delsarte-Hoffman Ratio bound and finally apply it to different graphs.
Minicourse 2: Wreath products and double coset graphs
Abstract: A graph Γ is vertex-transitive if, for any pair of vertices, there is an automorphism of Γ that maps one vertex to the other. In 1964 Sabidussi showed that every vertex-transitive graph is isomorphic to a double coset graph. So we may view the study of vertex-transitive graphs as the study of double coset graphs. At the same time, Sabidussi also explored the relationship between double coset graphs and Cayley graphs.
We will, in a self contained way, give the exact relationship between double coset graphs and Cayley graphs, which is done through the wreath product. We will also discuss consequences of this relationship, especially with regards to symmetry.