The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees

TL;DR

This paper characterizes the Laplacian polynomial of uniform hypergraphs, determines the multiplicity of zero eigenvalues in uniform hypertrees, and confirms a recent conjecture about their spectral properties.

Contribution

It provides a complete characterization of the zero Laplacian eigenvalue multiplicity in uniform hypertrees, confirming a recent conjecture and extending spectral theory of hypergraphs.

Findings

The zero Laplacian eigenvalue multiplicity in uniform hypertrees is explicitly determined.

The Laplacian characteristic polynomial for hypergraphs with cut vertices or pendant edges is characterized.

The conjecture on zero eigenvalue multiplicity in uniform hypertrees is proven.

Abstract

In this paper, the Laplacian characteristic polynomial of uniform hypergraphs with cut vertices or pendant edges and the Laplacian matching polynomial of uniform hypergraphs are characterized.The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees is given, which proves the conjecture in \cite{zheng2023zero} (The zero eigenvalue of the Laplacian tensor of a uniform hypergraph, Linear and Multilinear Algebra, (2023) Doi:10.1080/03081087.2023.2172541).