The multiplicity of eigenvalues of nonnegative weakly irreducible tensors and uniform hypergraphs

Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\mathcal{A}$ with an eigenvalue $\lambda$ and the corresponding eigenvariety ${\mathcal{V}}_{\lambda }(\mathcal{A})$, $$\mathrm{a}\mathrm{m}(\lambda )\ge \sum _{i=1}^{\kappa }\mathrm{d}\mathrm{i}\mathrm{m}(V_i)(k-1{)}^{\mathrm{d}\mathrm{i}\mathrm{m}(V_i)-1},$$ where $\mathrm{a}\mathrm{m}(\lambda )$ is the algebraic multiplicity of $\lambda$, and $V_1,\dots ,V_{\kappa }$ are all irreducible components of ${\mathcal{V}}_{\lambda }(\mathcal{A})$. In this paper, we prove that if $\mathcal{A}$ is a nonnegative weakly irreducible tensor with spectral radius $\rho$, then $\mathrm{a}\mathrm{m}(\lambda )\ge \mathrm{\mid }{\mathbb{V}}_{\lambda }(\mathcal{A})\mathrm{\mid }$ for all eigenvalues $\lambda$ of $\mathcal{A}$ with modulus $\rho$, where ${\mathbb{V}}_{\lambda }(\mathcal{A})$ is the projective eigenvariety of $\mathcal{A}$ associated with $\lambda$. Consequently we confirm Hu-Ye’s conjecture for the above eigenvalues $\lambda$ of $\mathcal{A}$ and also the least H-eigenvalue of a weakly irreducible $Z$-tensor. We prove several equality cases in Hu-Ye’s conjecture for the eigenvalues of the adjacency tensor or Laplacian tensor of uniform hypergraphs.