\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly ï¿½??-walk-regular with ï¿½??>1 if the number of walks of length ï¿½?? 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 ï¿½? with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of ï¿½?? for which ï¿½? can be strongly ï¿½??-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 ï¿½? for which there are infinitely many ï¿½?? for which ï¿½? is strongly ï¿½??-walk-regular.
\end{document}