The dimension of eigenvariety of nonnegative tensors associated with spectral radius

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar, called Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. We prove that the dimension of the above projective eigenvariety is zero, i.e. there are finitely many eigenvectors associated with the spectral radius up to a scalar. We characterize a nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs.