A Study of Hypergraph Using Null Spaces of the Incidence Matrix and its Transpose

TL;DR

This paper investigates hypergraph substructures through the null spaces of incidence matrices, revealing their spectral properties and implications for random walks and centrality measures.

Contribution

It introduces a novel spectral approach using null spaces of incidence matrices to characterize hypergraph substructures and their spectral and dynamic properties.

Findings

Null space vectors characterize hypergraph substructures.

Substructures show spectral similarities and redundancies.

Implications for random walks and centrality measures.

Abstract

In this study, we explore the substructures of a hypergraph that lead us to linearly dependent rows (or columns) in the incidence matrix of the hypergraph. These substructures are closely related to the spectra of various hypergraph matrices, including the signless Laplacian, adjacency, Laplacian, and adjacency matrices of the hypergraph's incidence graph. Specific eigenvectors of these hypergraph matrices serve to characterize these substructures. We show that vectors belonging to the nullspace of the adjacency matrix of the hypergraph's incidence graph provide a distinctive description of these substructures. Additionally, we illustrate that these substructures exhibit inherent similarities and redundancies, which manifest in analogous behaviours during random walks and similar values of hypergraph centralities.