The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph

In this paper, we show that the eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector associated with the zero eigenvalue of the Laplacian or signless Laplacian tensor have the same modulus. Moreover, under a canonical regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in {exp(2jπk)|ja∈[k]}. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with maximal support are called maximal. The maximal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and maximal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa.