We will continue on Marcus-Spielman-Srivastava's proof of the existence of infinite many Ramanujan graphs of any degree.
Their main result is: For a bipartite d-regular Ramanujan graph G, there exists a 2 lift of G which is also a bipartite d-regular Ramanujan Graph. The current goal is to show that convex combinations of signed adjacency matrices are real-rooted. The remaining of the proof will rely on multivariable real stable polynomials and linear operators that preserve real stable property. If time permits, we will discuss the proof of max root of a matching (defect) polynomial is bounded by 2\sqrt{d-1} for a d-regular graph. Speaker(s): Han Huang (University of Michigan)