Ramanujan Graphs in Polynomial Time

TL;DR

This paper presents a polynomial time algorithm for constructing bipartite Ramanujan graphs of all degrees and sizes, advancing from existence proofs to explicit computation.

Contribution

It introduces a deterministic polynomial time algorithm to explicitly compute bipartite Ramanujan graphs, building on prior existence results.

Findings

Polynomial time algorithm for bipartite Ramanujan graphs

Explicit construction of graphs of all degrees and sizes

Advancement from existence proofs to explicit algorithms

Abstract

The recent work by Marcus, Spielman and Srivastava proves the existence of bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that paper did not provide a polynomial time algorithm to actually compute such graphs. Here, we provide a polynomial time algorithm to compute certain expected characteristic polynomials related to this construction. This leads to a deterministic polynomial time algorithm to compute bipartite Ramanujan (multi)graphs of all degrees and all sizes.