# Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs

## TL;DR

This paper investigates the eigenvectors of Z-tensors related to their least H-eigenvalue, providing explicit counts for these eigenvectors in symmetric cases and establishing relationships among Laplacian, adjacency, and signless Laplacian eigenvectors in hypergraphs.

## Contribution

It characterizes the eigenvectors associated with the least H-eigenvalue of Z-tensors, especially in symmetric cases, and links these eigenvectors to hypergraph spectral properties.

## Key findings

- Finitely many eigenvectors associated with the least H-eigenvalue.
- Explicit count of eigenvectors via Smith normal form for symmetric Z-tensors.
- Equivalence of the number of Laplacian, adjacency, and signless Laplacian eigenvectors in hypergraphs.

## Abstract

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue.