# Directed strongly walk-regular graphs and their eigenvalues

### Edwin van Dam Tilburg University

#### Gholamreza Omidi Isfahan University of Technology

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Minisymposium: ASSOCIATION SCHEMES

Content: A directed graph is called strongly $\ell$-walk regular if the number of walks of length $\ell$ from one vertex to another depends only on whether the two vertices are the same, adjacent, or not adjacent. This generalizes the concept of directed strongly regular graphs and a problem introduced by Hoffman. It also generalizes the same concept for undirected graphs, which was studied by the authors in earlier work (JCTA 120 (2013), 803-810. arXiv:1208.3067). Here we present several results and constructions of directed strongly $\ell$-walk-regular graphs. Eigenvalue methods play a crucial role in obtaining these results.

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