Two conjectures in spectral hypergraph theory

Abstract

Let $\mathcal{A}$ be a $k$-th order $n$-dimensional tensor, and we denote by $\mathrm{am}(\hat{I}»,\mathcal{A})$ the algebraic multiplicity of the eigenvalue $\hat{I}»$ of $\mathcal{A}$. The projective eigenvariety ${\mathbb{V}}_{\hat{I}}»(\mathcal{A})$ is defined as the set of eigenvectors of $\mathcal{A}$ associated with $\hat{I}»$, 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 $\ddot{I}$ be the spectral radius of $\mathcal{A}(H)$, for which it is known that $|{\mathbb{V}}_{\ddot{I}}(\mathcal{A}(H))|=|{\mathbb{V}}_0(\mathcal{L}(H))|$. Recently, Fan [arXiv:2410.20830v2, 2024] conjectured that $\mathrm{am}(\ddot{I},\mathcal{A}(H))=|{\mathbb{V}}_{\ddot{I}}(\mathcal{A}(H))|$ and $\mathrm{am}(0,\mathcal{L}(H))=\mathrm{am}(\ddot{I},\mathcal{A}(H))$. In this paper, we prove these two conjectures, and thereby establish $$\mathrm{am}(\ddot{I},\mathcal{A}(H))=|{\mathbb{V}}_{\ddot{I}}(\mathcal{A}(H))|=|{\mathbb{V}}_0(\mathcal{L}(H))|=\mathrm{am}(0,\mathcal{L}(H)).$$ As shown by Fan et al., $|{\mathbb{V}}_{\ddot{I}}(\mathcal{A}(H))|$ 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.