Algebraic Graph Theory-Sebastian Cioabă-Spectral Moore theorems for graphs and hypergraphs

Abstract: The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander and Ramanujan graphs. In this talk, I will give an overview of recent work studying the maximum order v(k,\theta)  of a regular graph (bipartite graph or hypergraph) of given valency k whose second largest eigenvalue is at most a given value \theta. This problem can be seen as a spectral Moore problem and has connections to Alon-Boppana theorems for graphs and hypergraphs and with the usual Moore or degree-diameter problem.