Physical Sciences and Mathematics Commons^™

Coverings Of Graphs And Tiered Trees, Sam Glennon

Doctoral Dissertations

This dissertation will cover two separate topics. The first of these topics will be coverings of graphs. We will discuss a recent paper by Marcus, Spielman, and Srivastava proving the existence of infinite families of bipartite Ramanujan graphs for all regularities. The proof works by showing that for any d-regular Ramanujan graph, there exists an infinite tower of bipartite Ramanujan graphs in which each graph is a twofold covering of the previous one. Since twofold coverings of a graph correspond to ways of labeling the edges of the graph with elements of a group of order 2, we will generalize …