The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph

TL;DR

This paper explores the eigenvectors associated with zero Laplacian and signless Laplacian eigenvalues in uniform hypergraphs, revealing their relation to hypergraph components and phase distributions under regularization.

Contribution

It establishes the connection between eigenvectors and hypergraph components, characterizes their phases, and classifies eigenvectors into H- and N-types with minimal support.

Findings

Eigenvector components have the same modulus.

Phases are uniformly distributed under regularization.

Eigenvectors characterize bipartite and multipartite components.

Abstract

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a $k$-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a {\em canonical} regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in $\{\{exp}(\frac{2j\pi}{k})\;|\;j\in [k]\}$. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with minimal support is called {\em minimal}. The minimal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and the minimal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa.