Mustazee Rahman

On largest eigenvalues of graphs

Abstract: Classical Alon-Boppana Theorem provides a rather sharp lower bound on the second largest eigenvalue of a regular graph. Obtaining such bounds for non-regular graphs is more complicated. I will explain combinatorial and topological ideas that overcome some of these complications. It allows for Alon-Boppana type bounds for eigenvalues of large, bounded degree graphs. The bounds also apply to certain infinite, random graphs that are a stochastic generalization of finite graphs. This stochastic generalization, called unimodular networks, in fact plays a central role in deducing bounds for finite graphs.