It is a well known fact that the first eigenvalue of an n-vertex graph is at most n-1 while the second eigenvalue is at most n/2-1. Up until recently, even the question of bounding the third eigenvalue was open. In this talk, we will prove a general upper bound on the k-th eigenvalue of a graph, which is tight for k = 2,3,4,8 and 24. We will discuss the close relation of the graph eigenvalue problem with projection constants and equiangular lines.

Joint work with Quanyu Tang and Tanay Wakhare