  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 Laplaican (H- or N-)eigenvectors can be
get explicitly by solving the Smith normal form of the incidence matrix over
$\mathbb{Z}_m$ or $\mathbb{Z}_2$. 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 $G$.
