Some properties of Pareto $H$-eigenvalues on tensors and hypergraphs

Abstract

A Pareto $H$-eigenvalue of a tensor $\mathbb{A}$ is a real number $\lambda$ satisfying the complementarity system:
$0\leq x\perp (\mathbb{A}x-\lambda \mathbb{I} x)\geq 0$. The Pareto $H$-spectrum is the set of all Pareto $H$-eigenvalues. In this note, we first obtain some invariance on Pareto $H$-spectrum of tensors under tensor permutational similar (resp. diagonal similar) translation. As their application, we know that Pareto $H$-spectrum of hypergraphs is independent of the ordering of their vertices. Furthermore, we attain some nice properties on Pareto $H$-spectrum of hypergraphs. At the same time, we improve some results in \cite{Song} and \cite{Bo}.