Spectral directed hypergraph theory via tensors. Issue 4 (2nd April 2016) - Xie, Jinshan; Qi, Liqun
Abstract : 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.