New matrices for spectral hypergraph theory, II

TL;DR

This paper introduces three new hypergraph matrices—unified Laplacian, signless Laplacian, and normalized Laplacian—that connect hypergraph spectral properties with those of associated graphs, enabling structural analysis through eigenvalues.

Contribution

The paper defines three novel hypergraph matrices based on the unified matrix, linking hypergraph spectra to graph spectra and introducing related structures and invariants.

Findings

Unified hypergraph matrices are equivalent to classical graph matrices.

Spectral properties of these matrices reflect hypergraph structural features.

New invariants relate hypergraph structures to eigenvalues.

Abstract

The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian matrix, and unified normalized Laplacian matrix, all defined using the unified matrix. We show that these three matrices of a hypergraph are respectively identical to the Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the associated graph. This allows us to use the spectra of these hypergraph matrices as a means 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 these three matrices.