{
  "id": "2409.16055",
  "version": "v1",
  "published": "2024-09-24T12:58:18.000Z",
  "updated": "2024-09-24T12:58:18.000Z",
  "title": "On the Incidence matrices of hypergraphs",
  "authors": [
    "Samiron Parui"
  ],
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-24T12:58:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "15A03",
      "05C50",
      "05C07",
      "05C30"
    ],
    "keywords": [
      "hypergraph",
      "vertex-edge incidence matrix",
      "unit contraction",
      "null space",
      "edge-vertex incidence matrix remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}