Spectral properties of weighted line digraphs

In this paper, we treat some weighted line digraphs which are induced by a connected and undirected graph. For a given graph $G$, the adjacency matrix of the weighted line digraph $W$ is determined by a boundary operator from an arc-based space to a vertex-based space. We see that depending on the boundary operator and the Hilbert spaces, $W$ has different kind of an underlying stochastic transition operator. As an application, we obtain the spectrum of the positive support of cube of the Grover matrix in a large girth of the graph.