Using the Borcea Branden characterization of real stability preservers in the Weyl algebra, I shall describe a general strategy for proving that families of polynomials are real rooted. I shall carry out this strategy to prove a theorem due to Marcus, Spielman, and Srivastava that every graph has a Ramanujan 2 covering. I shall also outline some other proofs following the same idea.
Josiah Sugarman