Spectral directed hypergraph theory via tensors

In this paper, we show that each of the adjacency tensor, the Laplacian tensor and the signless Laplacian tensor of a uniform directed hypergraph has n linearly independent H-eigenvectors. Some lower and upper bounds for the largest and smallest adjacency, Laplacian and signless Laplacian H-eigenvalues of a uniform directed hypergraph are given. For a uniform directed hypergraph, the smallest Laplacian H-eigenvalue is 0. On the other hand, the upper bound of the largest adjacency and signless Laplacian H-eigenvalues are achieved if and only if it is a complete directed hypergraph. For a uniform directed hyperstar, all adjacency H-eigenvalues are 0. At the same time, we make some conjectures about the nonnegativity of one H-eigenvector corresponding to the largest H-eigenvalue, and raise some questions about whether the Laplacian and signless Laplacian tensors are positive semi-definite for a uniform directed hypergraph.