Directed strongly walk-regular graphs

We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\ell$-walk-regular with $\ell >1$ if the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. 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 $\Gamma$ with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of $\ell$ for which $\Gamma$ can be strongly $\ell$-walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with nonreal eigenvalues. We give such examples and characterize those digraphs $\Gamma$ for which there are infinitely many $\ell$ for which $\Gamma$ is strongly $\ell$-walk-regular

Directed strongly walk-regular graphs

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