Ramanujan Covering of Graphs
Ramanujan graphs are optimal expander graphs, and their existence and construction have been the focus of much research during the last three decades. We prove that every bipartite Ramanujan graph has a d-sheeted topological covering which is also Ramanujan. This generalizes the d=2 case, a recent major breakthrough in the subject due to Marcus, Spielman and Srivastava. The main tools we use are representations of groups and the Peter-Weyl theory, as well as the theory of interlacing polynomials. All notions will be explained.
Joint work with Chris Hall and Will Sawin.