2 research outputs found
Directed strongly walk-regular graphs
We generalize the concept of strong walk-regularity to directed graphs. We
call a digraph strongly -walk-regular with if the number of
walks of length 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 with only real eigenvalues whose adjacency matrix
is not diagonalizable has at most two values of for which can
be strongly -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 for
which there are infinitely many for which is strongly
-walk-regular