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 the content of the recent paper by discussing what happens when we label the edges of a graph by larger groups. We will give a version of their proof using threefold coverings instead of twofold coverings. We will also examine ways of reducing the size of the set of twofold coverings that we must consider when we follow the proof by Marcus, Spielman, and Srivastava.
The other topic that will be covered in this dissertation will be alternating trees and tiered trees. We will define a new generalization of alternating trees, which we will call tiered trees. We will also define a generalized weight system on these tiered trees. We will prove some enumerative results about tiered trees that demonstrate how they can be viewed as being obtained by applying certain procedures to certain types of alternating trees. We also provide a bijection between the set of permutations in Sn and the set of weight 0 alternating trees with n+1 vertices. We use this bijection to define a new statistic of permutations called the weight of a permutation, and use this weight to define a new q-Eulerian polynomial
