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13 - Ramanujan Graphs and Digraphs

Published online by Cambridge University Press:  14 August 2020

Matthias Keller
Affiliation:
Universität Potsdam, Germany
Daniel Lenz
Affiliation:
Universität Potsdam, Germany
Radoslaw K. Wojciechowski
Affiliation:
York College of the City University of New York
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Summary

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.

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Publisher: Cambridge University Press
Print publication year: 2020

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