Ramanujan Graphs and Digraphs

Ori Parzanchevski*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

2 Scopus citations


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.

Original languageAmerican English
Title of host publicationAnalysis and Geometry on Graphs and Manifolds
PublisherCambridge University Press
Number of pages24
ISBN (Electronic)9781108615259
ISBN (Print)9781108713184
StatePublished - 1 Jan 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© Cambridge University Press 2020.


  • Alon-Boppana
  • Cayley graphs
  • Directed graphs
  • Expanders
  • Ramanujan


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