Directed strongly walk-regular graphs

Abstract

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

A Simple Yet Effective SVD-GCN for Directed Graphs

“…Regardless of whether the graph is undirected or directed, as long as the adjacency matrix is diagonalizable, we can always factor A = VΛV −1 and generalize the Fourier transform as x = V −1 x. The adjacency matrix is diagonalizable for strongly connected directed graph [27] and we may consider Jordan decomposition when such condition is violated [20]. Notice that for general directed graphs, V −1 = V and also the S, V are complex-valued, which poses difficulty to extend classic wavelet/framelet theories.…”