Ramanujan Covers of Graphs

Chris Hall, University of Western Ontario

Given a k-regular (undirected) graph G with n vertices, we will define what it means for G to be Ramanujan.  It implies that an associated zeta function satisfies a Riemann hypothesis.  It also implies that a sequence G_1,G_2,... of Ramanujan k-regular graphs where the number of vertices n_i tends to infinity, will be expander with optimal expansion constant.  A spectacular result of Marcus, Spielman, and Srivastava implies there exist *infinitely* many Ramanujan k-regular graphs for *every* k.  We will describe this result and a refinement due to D. Puder, W. Sawin, and the speaker.

MC 5501