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New matrices for spectral hypergraph theory, I

R. Vishnupriya, R. Rajkumar

Abstract

We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to use the spectrum of the unified matrix of a hypergraph as a tool to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of the unified matrix.

New matrices for spectral hypergraph theory, I

Abstract

We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to use the spectrum of the unified matrix of a hypergraph as a tool to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of the unified matrix.

Paper Structure

This paper contains 12 sections, 30 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unified matrix of a hypergraph
  4. Basic results on the unified matrix of a hypergraph
  5. New structures and invariants of hypergraphs and unified eigenvalues
  6. Elementary $q$-subhypergraphs and the determinant of the unified matrix of a hypergraph
  7. Unified spectrum of unified cycles and unified paths
  8. Unified characteristic polynomial of the hypergraphs constructed by some hypergraph operations
  9. Bounds
  10. Unified spectral radius and some hypergraph invariants
  11. Chromatic numbers and unified eigenvalues of a hypergraph
  12. Independence number, complete-clique number and unified eigenvalues of a hypergraph

Key Result

Lemma 4.1

Let $H$ be a hypergraph with $e$-index $k$ and let $\mathbf{U}(H)=(\mathbf{U}_{S_iS_j})_{1\leq i,j\leq k}$ be its unified matrix. Then we have the following.

Figures (5)

  • Figure 1: The hypergraph $H$
  • Figure 2: The associated graph $G_{H'}$ of $H'$
  • Figure 3: An exactly connected hypergraph $H$
  • Figure 5: The hypergraph $H_2$
  • Figure 6: The hypergraph $H$ and its elementary $10$-subhypergraphs $H_1$, $H_2$, $H_3$.

Theorems & Definitions (82)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Example 3.1
  • Example 3.2
  • Definition 4.1
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 72 more