The multiplicity of eigenvalues of nonnegative tensors and hypergraphs

Hu and Ye conjectured that for an $n$-dimensional tensor $\mathcal{A}$ of order $k$ with an eigenvalue $\lambda$ and the corresponding eigenvariety $\mathcal{V}_\lambda(\mathcal{A})$, the algebraic multiplicity $\mathrm{am}(\lambda)$ of $\lambda$ satisfies: $$\mathrm{am}(\lambda) \ge \sum_{i=1}^\kappa \dim(V_i)(k-1)^{\dim(V_i)-1},$$ where $V_1,\ldots,V_\kappa$ are all irreducible components of $\mathcal{V}_\lambda(\mathcal{A})$. In this paper, we establish that for any weakly irreducible nonnegative tensor $\mathcal{A}$ with spectral radius $\rho$, all the eigenvalues $\lambda$ of $\mathcal{A}$ with $|\lambda|=\rho$ satisfy $\mathrm{am}(\lambda) \ge |\mathbb{V}_\lambda(\mathcal{A})|$, where $\mathbb{V}_\lambda(\mathcal{A})$ is the projective eigenvariety of $\mathcal{A}$ associated with $\lambda$. As a direct consequence, we confirm the Hu-Ye Conjecture for two classes of eigenvalues: (1) all eigenvalues of weakly irreducible nonnegative tensors with modulus equal to the spectral radius, and (2) the least H-eigenvalues of weakly irreducible $Z$-tensors. Furthermore, we characterize the equality condition in Hu-Ye's conjecture for the eigenvalues of several hypergraph classes and present a new conjecture regarding the eigenvalue multiplicities of hypergraphs. As an initial step toward this new conjecture, we give a sufficient condition for a point in the projective eigenvariety of a tensor to have local intersection multiplicity one.