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On the Incidence matrices of hypergraphs

Samiron Parui

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.

On the Incidence matrices of hypergraphs

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 and vertex , the -th entry of the matrix is if is incident to ; otherwise, this entry is . 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 -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.
Paper Structure (4 sections, 23 theorems, 9 equations, 2 figures)

This paper contains 4 sections, 23 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The null spaces of Incidence matrices of a hypergraph
  3. Incidence matrix and eigenvalues of other Hypergraph matrices
  4. Conclusion

Key Result

Proposition 2.4

For some natural number $k>1$, if $n$ is a multiple of $k$, then $B_{C_n^k}x_{\omega}=0$, where $\omega(\ne 1)$ is a $k$-th root of unity.

Figures (2)

  • Figure 1: A hypergraph $H$ with $V(H) = \{n \in \mathbb{N} : n \leq 8\}$ and $E(H) = \{e_1 = \{1, 2, 3, 4, 7\}, e_2 = \{2, 3, 4, 5, 8\}, e_3 = \{3, 4, 5, 6\}, e_4 = \{4, 5, 6, 1\}, e_5 = \{5, 6, 1, 2\}, e_6 = \{6, 1, 2, 3\}\}$. The subset $U = \{n \in \mathbb{N} : n \leq 6\} \subset V(H)$ of the vertex set the induced the sub-hypergraph $H_U=C_6^4$. The pairwise-disjoint collection of vertices $W=\{1,3,5\}$, $U=\{2,4\}$, and $V=\{7,8\}$ are such that $(|e\cap U|-|e\cap V|):(e\cap W)=\frac{1}{2}$ for all $e\in E(H)$.
  • Figure 2: Units and unit contraction of a hypergraph $H$ with $V(H)=\{1,2,\ldots,10,11\}$, and $E(H)=\{ e_1=\{1,2,5,6,7,10,11,\},e_2=\{1,2,3,4\}, e_3=\{3,4,10\},e_4=\{5,6,7,8,9\},e_5=\{8,9,10,11\}\}$.

Theorems & Definitions (60)

  • Definition 2.2: $k$-uniform cycle
  • Example 2.3: $4$-uniform cycle of length $8$, $C_8^4$
  • Proposition 2.4
  • proof
  • Example 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Theorem 2.8
  • proof
  • ...and 50 more