The multiplicity of eigenvalues of nonnegative tensors and hypergraphs

Abstract

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 $V_1,\dots ,V_{\kappa }$ 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.