H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors

TL;DR

This paper introduces a natural definition for Laplacian and signless Laplacian tensors of uniform hypergraphs, analyzing their H-eigenvalues and establishing key properties and applications in connectivity.

Contribution

It provides a new framework for defining and studying Laplacian tensors of hypergraphs, including eigenvalue properties and connectivity measures.

Findings

Each tensor has at most one positive eigenvalue.

Largest and smallest H^+-eigenvalues are identified.

Connections to edge connectivity are established.

Abstract

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H$^+$-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H$^{++}$-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at most one H$^{++}$-eigenvalue, but has several other H$^+$-eigenvalues. We identify their largest and smallest H$^+$-eigenvalues, and establish some maximum and minimum properties of these H$^+$-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.