Existence and polynomial time construction of biregular, bipartite Ramanujan graphs of all degrees

TL;DR

This paper proves the existence and provides a polynomial-time construction method for bipartite, biregular Ramanujan graphs of all degrees and sizes under certain divisibility conditions, extending previous results.

Contribution

It generalizes the existence and construction of biregular Ramanujan graphs to all degrees and sizes with a new proof technique involving rectangular convolutions.

Findings

Existence of biregular Ramanujan graphs for all degrees and sizes under divisibility conditions.

Polynomial-time algorithm for constructing such graphs.

Extension of previous work to biregular cases.

Abstract

We prove that there exist bipartite, biregular Ramanujan graphs of every degree and every number of vertices provided that the cardinalities of the two sets of the bipartition divide each other. This generalizes a result of Marcus, Spielman, and Srivastava and, similar to theirs, the proof is based on the analysis of expected polynomials. The primary difference is the use of some new machinery involving rectangular convolutions, developed in a companion paper. We also prove the constructibility of such graphs in polynomial time in the number of vertices, extending a result of Cohen to this biregular case.