New matrices for spectral hypergraph theory, I

TL;DR

This paper introduces a new hypergraph matrix called the unified matrix, which allows hypergraphs to be represented as graphs, enabling spectral analysis to connect hypergraph structure with graph properties.

Contribution

The paper presents the unified matrix for hypergraphs, establishing a spectral framework that links hypergraph invariants to eigenvalues of the associated matrix.

Findings

Unified matrix is identical to the adjacency matrix of the associated graph.

Spectral properties of the unified matrix reflect hypergraph structural invariants.

New hypergraph structures are related to eigenvalues of the unified matrix.

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.