On the second-largest modulus among the eigenvalues of a power hypergraph

TL;DR

This work studies the second-largest eigenvalue modulus $\Lambda$ of the $k$-power hypergraph $G^{(k)}$ and ties it to the weakest edges of the base graph $G$. By leveraging signed subgraphs and the projective eigenvariety framework, it obtains a sharp formula $\Lambda^k=\rho_E(G)^2$ for $k\ge4$ (with a complete case analysis for $k=3$) and proves that $\mathbb{V}_{\Lambda}(G^{(k)})$ is zero-dimensional, i.e., there are finitely many eigenvectors up to scalar. The authors then give explicit expressions for the algebraic multiplicity $\mathrm{am}_{\Lambda}(G^{(k)})$ and the total multiplicity $\#\mathbb{V}_{\Lambda}(G^{(k)})}$, showing $\mathrm{am}_{\Lambda}(G^{(k)})=\#\mathbb{V}_{\Lambda}(G^{(k)})$ and representing both in terms of the number of weakest edges. These results crystallize how edge fragility in $G$ governs the secondary spectral structure of its power hypergraph and connect hypergraph tensor spectra to signed-subgraph dynamics.

Abstract

It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph $G=(V,E)$ and $k \geq 3$, the $k$-power hypergraph $G^{(k)}$ is a $k$-uniform hypergraph obtained by adding $k-2$ new vertices to each edge of $G$, who always has non-real eigenvalues. In this paper, we determine the second-largest modulus $Λ$ among the eigenvalues of $G^{(k)}$, which is indeed an eigenvalue of $G^{(k)}$. The projective eigenvariety $\mathbb{V}_Λ$ associated with $Λ$ is the set of the eigenvectors of $G^{(k)}$ corresponding to $Λ$ considered in the complex projective space. We show that the dimension of $\mathbb{V}_Λ$ is zero, i.e, there are finitely many eigenvectors corresponding to $Λ$ up to a scalar. We give both the algebraic multiplicity of $Λ$ and the total multiplicity of the eigenvector in $\mathbb{V}_Λ$ in terms of the number of the weakest edges of $G$. Our result show that these two multiplicities are equal.

Figures (1)

• Figure .1: Examples for Theorem \ref{['thm4']}

Theorems & Definitions (29)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8