Some spectral properties of uniform hypergraphs

TL;DR

This paper explores spectral properties of uniform hypergraphs, deriving trace formulas for the Laplacian tensor, characterizations of hypergraph classes, and verifying conjectures under specific conditions.

Contribution

It introduces new trace formulas for the Laplacian tensor, spectral characterizations of hypergraph classes, and partial answers to existing conjectures.

Findings

Sum of degree powers determined by Laplacian spectrum

Spectral characterizations of odd-bipartite hypergraphs

Verification of a conjecture under certain conditions

Abstract

For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree sequence of $H$. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al \cite{ShaoShanWu}. We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al \cite{HuQiShao} holds under certain conditons.