Spectral Properties of Oriented Hypergraphs

TL;DR

This paper investigates the spectral properties of oriented hypergraphs, establishing eigenvalue bounds, exploring duality, and generalizing graph spectral results to hypergraphs with signed incidences.

Contribution

It introduces eigenvalue bounds for adjacency and Laplacian matrices of oriented hypergraphs and extends spectral graph theory concepts to hypergraph settings.

Findings

Eigenvalue bounds depend on structural parameters.

Oriented hypergraph and its incidence dual share nonzero Laplacian eigenvalues.

A family of hypergraphs generalizes the signless Laplacian of graphs.

Abstract

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.