Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue

TL;DR

This paper investigates the eigenvectors associated with zero eigenvalues of Laplacian and signless Laplacian tensors in hypergraphs, providing explicit formulas and connections to bipartitions.

Contribution

It explicitly determines the number of such eigenvectors using the incidence matrix and establishes their relation to hypergraph bipartitions.

Findings

Number of eigenvectors equals for Laplacian and signless Laplacian when zero eigenvalue occurs.

Explicit formulas for counting eigenvectors via Smith normal form.

Connection between eigenvectors and bipartitions of the hypergraph.

Abstract

Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over $\mathbb{Z}_m$ or $\mathbb{Z}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of $G$.