Eigenvectors of Laplacian or signless Laplacian of hypergraphs associated with zero eigenvalue

Abstract Let G be a connected m -uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of G associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of G . By means of the incidence matrix of G , the number of first Laplacian or signless Laplacian (or H-)eigenvectors can be obtained explicitly by solving the Smith normal form of the incidence matrix over Z m (or Z 2 ). Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of G .