### Abstract

Original language | English |
---|---|

Pages (from-to) | 623–639 |

Journal | Journal of Algebraic Combinatorics |

Volume | 47 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jun 2018 |

### Fingerprint

### Keywords

- Strongly regular digraph
- Walk
- Spectrum
- Eigenvalues

### Cite this

*Journal of Algebraic Combinatorics*,

*47*(4), 623–639. https://doi.org/10.1007/s10801-017-0789-8

}

*Journal of Algebraic Combinatorics*, vol. 47, no. 4, pp. 623–639. https://doi.org/10.1007/s10801-017-0789-8

**Directed strongly walk-regular graphs.** / van Dam, Edwin; Omidi, G.R.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Directed strongly walk-regular graphs

AU - van Dam, Edwin

AU - Omidi, G.R.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Strongly regular digraph

KW - Walk

KW - Spectrum

KW - Eigenvalues

U2 - 10.1007/s10801-017-0789-8

DO - 10.1007/s10801-017-0789-8

M3 - Article

VL - 47

SP - 623

EP - 639

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 4

ER -