Ramanujan graphs and digraphs

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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 languageEnglish
Title of host publicationAnalysis and geometry on graphs and manifolds
Subtitle of host publicationSelected papers of the conference, University of Potsdam, Potsdam, Germany, July 31 -- August 4, 2017
PublisherCambridge University Press
Number of pages24
ISBN (Print)978-1-108-71318-4; 978-1-108-61525-9
StatePublished - 2020

Publication series

NameLondon Mathematical Society Lecture Note Series
PublisherCambridge University Press


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


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