All finite transitive graphs admit a self-adjoint free semigroupoid algebra

Adam Dor-On, Christopher Linden

Research output: Contribution to journalArticlepeer-review


In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is. This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Colouring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.

Original languageEnglish
Pages (from-to)391-406
Number of pages16
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number1
StatePublished - Feb 2021
Externally publishedYes

Bibliographical note

Publisher Copyright:
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.


  • Cuntz Krieger
  • Cyclic decomposition
  • Directed graphs
  • Free semigroupoid algebra
  • Graph algebra
  • Periodic
  • Road colouring

ASJC Scopus subject areas

  • General Mathematics


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