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Line Multigraphs of Hypergraphs

Kauê Cardoso

TL;DR

This work introduces line multigraphs as a bridge between hypergraph incidence and graph spectral theory by representing hyperedges as vertices and encoding pairwise intersections with edge multiplicities. It establishes that the line multigraph preserves key structural properties via skew edge-regularity, derives a fundamental spectral bound $\lambda \ge -r$ for rank $r$, and identifies collars as a mechanism to attain the bound. The authors connect these structural and spectral insights to the signless Laplacian of hypergraphs, providing bounds and spectrum characterizations for general and power hypergraphs, and show that power hypergraphs induce precise eigenvalue shifts through line-multigraph scaling. Overall, the paper offers a unified, tractable framework to extend classical line-graph results to general hypergraphs, with concrete implications for spectral hypergraph theory and related applications.

Abstract

A line multigraph is obtained from a hypergraph as follows: the vertices of the multigraph correspond to the hyperedges of the hypergraph, and the number of edges between two vertices is given by the number of vertices shared by the corresponding hyperedges. In this paper, we establish several structural and spectral properties of this class of multigraphs. More precisely, we show that important structural characteristics, such as connectivity, linearity, and regularity are, in some sense, preserved between a hypergraph and its line multigraph. We also prove that the eigenvalues of the line multigraph associated with a general hypergraph of rank $r$ are greater than or equal to $-r$, which generalizes a fundamental spectral property of line graphs. Furthermore, we provide sufficient conditions for $-r$ to be an eigenvalue of the line multigraph. Finally, we present applications of line multigraphs to the spectral theory of hypergraphs, including bounds for the signless Laplacian spectral radius of a hypergraph and a characterization of the signless Laplacian spectrum for a specific class of hypergraphs.

Line Multigraphs of Hypergraphs

TL;DR

This work introduces line multigraphs as a bridge between hypergraph incidence and graph spectral theory by representing hyperedges as vertices and encoding pairwise intersections with edge multiplicities. It establishes that the line multigraph preserves key structural properties via skew edge-regularity, derives a fundamental spectral bound for rank , and identifies collars as a mechanism to attain the bound. The authors connect these structural and spectral insights to the signless Laplacian of hypergraphs, providing bounds and spectrum characterizations for general and power hypergraphs, and show that power hypergraphs induce precise eigenvalue shifts through line-multigraph scaling. Overall, the paper offers a unified, tractable framework to extend classical line-graph results to general hypergraphs, with concrete implications for spectral hypergraph theory and related applications.

Abstract

A line multigraph is obtained from a hypergraph as follows: the vertices of the multigraph correspond to the hyperedges of the hypergraph, and the number of edges between two vertices is given by the number of vertices shared by the corresponding hyperedges. In this paper, we establish several structural and spectral properties of this class of multigraphs. More precisely, we show that important structural characteristics, such as connectivity, linearity, and regularity are, in some sense, preserved between a hypergraph and its line multigraph. We also prove that the eigenvalues of the line multigraph associated with a general hypergraph of rank are greater than or equal to , which generalizes a fundamental spectral property of line graphs. Furthermore, we provide sufficient conditions for to be an eigenvalue of the line multigraph. Finally, we present applications of line multigraphs to the spectral theory of hypergraphs, including bounds for the signless Laplacian spectral radius of a hypergraph and a characterization of the signless Laplacian spectrum for a specific class of hypergraphs.
Paper Structure (6 sections, 21 theorems, 26 equations, 3 figures)

This paper contains 6 sections, 21 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural properties of line multigraphs
  4. Spectral properties of line multigraphs
  5. Applications to Spectral Hypergraph Theory
  6. Conclusion

Key Result

Lemma 2.6

Let $\mathcal{H}$ be a hypergraph without isolated vertices. Then, $\mathcal{L}(\mathcal{H})$ is connected if and only if $\mathcal{H}$ is connected.

Figures (3)

  • Figure 1: The hypergraph $\mathcal{H}$ and its line multigraph $\mathcal{L}(\mathcal{H})$.
  • Figure 2: A $3$-uniform collar.
  • Figure 3: The power hypergraph $(P_4)^5_2$.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 57 more