Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

TL;DR

This paper proves the existence of bipartite Ramanujan graphs of all sizes and degrees by analyzing expected characteristic polynomials and employing interlacing families and finite free convolutions.

Contribution

It introduces a novel method to construct bipartite Ramanujan graphs of any size and degree using advanced polynomial analysis techniques.

Findings

Existence of bipartite Ramanujan graphs for all degrees and sizes.

Development of a new framework involving expected characteristic polynomials.

Application of finite free convolutions to bound polynomial roots.

Abstract

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced.