Ramanujan Graphs and Digraphs

TL;DR

This paper investigates Ramanujan digraphs, extending classical graph theory concepts to directed graphs, and demonstrates their unique spectral and combinatorial properties, including their construction, spectral bounds, and applications.

Contribution

It introduces the concept of Ramanujan digraphs, explores their properties, and provides new results on their construction and spectral characteristics.

Findings

Almost-normal Ramanujan digraphs are extremal in Alon-Boppana sense

They exhibit small diameter and optimal covering time

Connections to Cayley graphs and explicit constructions are established

Abstract

Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown to be of special interest, as they are extreme in the sense of an Alon-Boppana theorem, and they have remarkable combinatorial features, such as small diameter, Chernoff bound for sampling, optimal covering time and sharp cutoff. Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alon's conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan digraphs.