Two conjectures in spectral hypergraph theory

Let $\mathcal{A}$ be a $k$-th order $n$-dimensional tensor, and we denote by ${\rm am}(\lambda, \mathcal{A})$ the algebraic multiplicity of the eigenvalue $\lambda$ of $\mathcal{A}$. The projective eigenvariety $\mathbb{V}_\lambda(\mathcal{A})$ is defined as the set of eigenvectors of $\mathcal{A}$ associated with $\lambda$, considered in the complex projective space. For a connected uniform hypergraph $H$, let $\mathcal{A}(H)$ and $\mathcal{L}(H)$ denote its adjacency tensor and Laplacian tensor, respectively. Let $\rho$ be the spectral radius of $\mathcal{A}(H)$, for which it is known that $|\mathbb{V}_{\rho}(\mathcal{A}(H))| = |\mathbb{V}_{0}(\mathcal{L}(H))|$. Recently, Fan [arXiv:2410.20830v2, 2024] conjectured that ${\rm am}(\rho, \mathcal{A}(H)) = |\mathbb{V}_{\rho}(\mathcal{A}(H))|$ and ${\rm am}(0, \mathcal{L}(H)) = {\rm am}(\rho, \mathcal{A}(H))$. In this paper, we prove these two conjectures, and thereby establish $${\rm am}(\rho, \mathcal{A}(H)) = |\mathbb{V}_{\rho}(\mathcal{A}(H))| = |\mathbb{V}_{0}(\mathcal{L}(H))| = {\rm am}(0, \mathcal{L}(H)).$$ As shown by Fan et al., $|\mathbb{V}_{\rho}(\mathcal{A}(H))|$ and $|\mathbb{V}_{0}(\mathcal{L}(H))|$ can be computed via the Smith normal form of the incidence matrix of $H$ over $\mathbb{Z}_{k}$. Consequently, we provide a method for computing the algebraic multiplicity of the spectral radius and zero Laplacian eigenvalue for connected uniform hypergraphs.