Interlacing Families and Ramanujan Graphs

This lecture covers the concept of interlacing families of polynomials, focusing on 1-sided Ramanujan graphs. The instructor explains the properties of these graphs, such as d-regularity and eigenvalues. The lecture delves into two approaches for constructing interlacing families, including the use of matching polynomials. The connection between characteristic polynomials and matching polynomials in trees and graphs is explored, along with theorems by Godsil and Gutman. The lecture concludes with discussions on weighted adjacency matrices and the significance of real roots in matching polynomials.