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 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 \(\varGamma \) with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of \(\ell \) for which \(\varGamma \) can be strongly \(\ell \)-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 \(\varGamma \) for which there are infinitely many \(\ell \) for which \(\varGamma \) is strongly \(\ell \)-walk-regular.