The geometry connectivity of hypergraphs

TL;DR

This paper introduces the concept of geometry connectivity in hypergraphs, showing it equals the number of connected components, thus linking spectral properties of the Laplacian tensor to hypergraph connectivity.

Contribution

It defines the geometry connectivity of a hypergraph and proves its equivalence to the number of connected components, providing a spectral characterization of hypergraph connectivity.

Findings

Geometry connectivity equals the number of connected components.

Spectral properties of the Laplacian tensor characterize hypergraph connectivity.

Provides a new spectral tool for analyzing hypergraph structure.

Abstract

Let $\mathcal{G}$ be a $k$-uniform hypergraph, $\mathcal{L}_{\mathcal{G}}$ be its Laplacian tensor. And $\beta( \mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $\mathcal{L}_{\mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $\beta( \mathcal{G})$ is called the geometry connectivity of $\mathcal{G}$. We show that the number of connected components of $\mathcal{G}$ equals the geometry connectivity $\beta( \mathcal{G})$.