Two conjectures in spectral hypergraph theory

TL;DR

This paper proves two conjectures linking algebraic multiplicities and eigenvarieties in spectral hypergraph theory, providing a method to compute these multiplicities for connected uniform hypergraphs.

Contribution

It confirms Fan's conjectures, establishing equalities between algebraic multiplicities and eigenvariety sizes, and offers a computational method based on the Smith normal form.

Findings

Proved that algebraic multiplicity of spectral radius equals the size of the eigenvariety.

Established that algebraic multiplicity of zero Laplacian eigenvalue equals the spectral radius multiplicity.

Provided a method to compute these multiplicities using the Smith normal form.

Abstract

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.