NUMBER-THEORETIC CONSTRUCTIONS OF RAMANUJAN GRAPHS

A Ramanujan graph is a k-regular graph with nontrivial eigenvalues bounded by 2 root k - 1. In this article we review three explicit constructions of Ramanujan graphs using results from automorphic forms, representation theory of GL(2) over finite fields, and character sum estimates derived from the Riemann Hypothesis for curves over finite fields.