Some new trace formulas of tensors with applications in spectral hypergraph theory

TL;DR

This paper introduces new trace formulas for tensors that facilitate spectral analysis of hypergraphs, providing characterizations of spectral symmetry and relations between Laplacian and signless Laplacian spectra.

Contribution

It presents novel trace formulas for tensors that do not rely on differential operators, enabling new spectral characterizations of hypergraphs.

Findings

Characterization of k-uniform hypergraphs with k-symmetric spectra

Generalization of previous results on spectral symmetry

Simplified proof for trace expressions of tensors

Abstract

We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ of a tensor $\mathbb {T}$ which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the $k$-uniform hypergraphs whose spectra are $k$-symmetric, thus give an answer to a question raised in [3]. We generalize the results in [3, Theorem 4.2] and [5, Proposition 3.1] about the $k$-symmetry of the spectrum of a $k$-uniform hypergraph, and answer a question in [5] about the relation between the Laplacian and signless Laplacian spectra of a $k$-uniform hypergraph when $k$ is odd. We also give a simplified proof of an expression for $Tr_2(\mathbb {T})$ and discuss the expression for $Tr_3(\mathbb {T})$.