Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

TL;DR

This paper proves the existence of infinite families of bipartite Ramanujan graphs for all degrees greater than 2, introducing a new interlacing polynomials method to establish these combinatorial structures.

Contribution

It introduces the method of interlacing polynomials and proves the existence of bipartite Ramanujan graphs of all degrees greater than 2, including irregular variants.

Findings

Existence of infinite bipartite Ramanujan graphs for all degrees > 2

Construction of irregular Ramanujan graphs with eigenvalues bounded by spectral radius

Development of the interlacing polynomials technique

Abstract

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of `irregular Ramanujan' graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c,d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by sqrt{c-1}+sqrt{d-1}, for all c, d \geq 3. Our proof exploits a new technique for demonstrating the existence of useful combinatorial objects that we call the "method of interlacing polynomials'".