Directed strongly walk-regular graphs

Edwin van Dam, G.R. Omidi

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly ℓ -walk-regular with ℓ>1 if the number of walks of length ℓ from a vertex to another vertex depends only on whether the first vertex is the same as, adjacent to, or not adjacent to the second vertex. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph Γ with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of ℓ for which Γ can be strongly ℓ -walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with non-real eigenvalues. We give such examples and characterize those digraphs Γ for which there are infinitely many ℓ for which Γ is strongly ℓ -walk-regular.
Original languageEnglish
Pages (from-to)623–639
JournalJournal of Algebraic Combinatorics
Volume47
Issue number4
DOIs
Publication statusPublished - 1 Jun 2018

Keywords

  • Strongly regular digraph
  • Walk
  • Spectrum
  • Eigenvalues

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