  Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor
$\mathcal{A}$ can have more than one eigenvector associated with the least
H-eigenvalue. We show that there are finitely many eigenvectors of
$\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is
further combinatorial symmetric, the number of such eigenvectors can be
obtained explicitly by the Smith normal form of the incidence matrix of
$\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove
that the number of Laplacian eigenvectors of $G$ associated with the zero
eigenvalue is equal to the the number of adjacency eigenvectors of $G$
associated with the spectral radius, which is also equal to the number of
signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if
zero is an signless Laplacian eigenvalue.
