On the Incidence matrices of hypergraphs

TL;DR

This paper investigates the properties of incidence matrices in hypergraphs, analyzing their ranks, null spaces, and relationships with hypergraph structures like units and cycles, and explores their connection to eigenvalues of adjacency matrices.

Contribution

It provides new insights into the rank and null space of hypergraph incidence matrices and their invariance under unit contraction, linking matrix properties to hypergraph structures.

Findings

Rank of incidence matrices is preserved under unit contraction.

Certain hypergraph structures influence null space vectors.

Connections established between incidence matrices and adjacency matrix eigenvalues.

Abstract

This study delves into the incidence matrices of hypergraphs, with a focus on two types: the edge-vertex incidence matrix and the vertex-edge incidence matrix. The edge-vertex incidence matrix is a matrix in which the rows represent hyperedges and the columns represent vertices. For a given hyperedge $e$ and vertex $u$, the $(e,u)$-th entry of the matrix is $1$ if $u$ is incident to $e$; otherwise, this entry is $0$. The vertex-edge incidence matrix is simply the transpose of the edge-vertex incidence matrix. This study examines the ranks and null spaces of these incidence matrices. It is shown that certain hypergraph structures, such as $k$-uniform cycles, units, and equal partitions of hyperedges and vertices, can influence specific vectors in the null space. In a hypergraph, a unit is a maximal collection of vertices that are incident with the same set of hyperedges. Identification of vertices within the same unit leads to a smaller hypergraph, known as unit contraction. The rank of the edge-vertex incidence matrix remains the same for both the original hypergraph and its unit contraction. Additionally, this study establishes connections between the edge-vertex incidence matrix and certain eigenvalues of the adjacency matrix of the hypergraph.