UC San Diego

In Chapter I, a brief history of expander graphs will be discussed. In Chapter II, I will introduce many elementary terms and concepts related to graphs and graph covers. Subsequently, in Chapter III, I will study logarithmic derivatives of L-functions associated to graph covers. In this chapter I will show how to use the representations associated to a graph covering, to determine the number of paths which split completely in a given cover. In Chapter IV, an explicit formula for graph zeta functions will be presented. Subsequently I will combine elements of the previous chapter to deduce an explicit formula for graph L -functions. In the next chapter, the subject of the universal cover of a graph and its spectrum will be discussed. A result of Angel, Friedman, and Hoory will be discussed. I will prove a theorem allowing one to increase the speed of their result. I will use this to apply the aforementioned result to a new graph ; this will allow us to provide new evidence of a conjecture they provide in their paper [3]