Two conjectures in spectral hypergraph theory
Ya-Nan Zheng
TL;DR
The paper settles Fan's conjectures by linking algebraic multiplicities of spectral hypergraph tensors to sizes of projective eigenvarieties. It achieves this through a reduction to stochastic form via diagonal similarity, then employs Macaulay's resultant and Hilbert function machinery to show that the algebraic multiplicity of the spectral radius equals the number of eigenvectors in the corresponding projective variety, and that this equals the multiplicity of the zero Laplacian eigenvalue. This yields a computable framework: the Smith normal form of incidence matrices over $Z_k$ determines both the eigenvariety sizes and the multiplicities. The results extend to a variety of hypergraph classes and operations, providing explicit formulas and multiplicativity properties that enable practical calculation of these spectral invariants, with further simplifications when $k$ is prime.
Abstract
Let $\mathcal{A}$ be a $k$-th order $n$-dimensional tensor, and we denote by ${\rm am}(λ, \mathcal{A})$ the algebraic multiplicity of the eigenvalue $λ$ of $\mathcal{A}$. The projective eigenvariety $\mathbb{V}_λ(\mathcal{A})$ is defined as the set of eigenvectors of $\mathcal{A}$ associated with $λ$, 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 $ρ$ be the spectral radius of $\mathcal{A}(H)$, for which it is known that $|\mathbb{V}_ρ(\mathcal{A}(H))| = |\mathbb{V}_{0}(\mathcal{L}(H))|$. Recently, Fan [arXiv:2410.20830v2, 2024] conjectured that ${\rm am}(ρ, \mathcal{A}(H)) = |\mathbb{V}_ρ(\mathcal{A}(H))|$ and ${\rm am}(0, \mathcal{L}(H)) = {\rm am}(ρ, \mathcal{A}(H))$. In this paper, we prove these two conjectures, and thereby establish $$ {\rm am}(ρ, \mathcal{A}(H)) = |\mathbb{V}_ρ(\mathcal{A}(H))| = |\mathbb{V}_{0}(\mathcal{L}(H))| = {\rm am}(0, \mathcal{L}(H)). $$ As shown by Fan et al., $|\mathbb{V}_ρ(\mathcal{A}(H))|$ 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.