Generalizing Lattice Structures to Hypergraphs: Spectra of Clique and Hyperedge-based Laplacians

TL;DR

This paper extends the concept of lattice structures from graphs to hypergraphs, analyzing their Laplacian spectra through two definitions, revealing how higher-order interactions influence spectral properties.

Contribution

It introduces a generalized lattice framework for hypergraphs and derives explicit spectral characterizations for two types of hypergraph Laplacians, advancing spectral hypergraph theory.

Findings

Derived Laplacian matrices for hypergraph lattices.

Analyzed eigenvalue spectra in relation to hyperedge size and lattice parameters.

Provided explicit spectral formulas capturing higher-order interaction effects.

Abstract

Lattice structures play a central role in spectral graph theory, offering analytical insight into diffusion, synchronization, and transport processes on regular discrete spaces. While their spectral properties are completely characterized in the classical graph setting, an extension to hypergraphs, where interactions involve more than two nodes, remains largely unexplored in the matrix-based formulation. In this work, we generalize the notion of a lattice to the hypergraph framework and study its Laplacian spectra under two alternative definitions: the clique Laplacian, obtained through pairwise projection, and the hyperedge-based Laplacian, defined via normalized hyperedge incidences. For both definitions, we derive the corresponding Laplacian matrices, analyze their eigenvalue spectra, and discuss how they reflect the underlying topological and dynamical structure of the hyperlattice. Our main result is a theorem giving a full spectral characterization in the periodic case, together with a Toeplitz-type open analogue whose spectrum retains a separable trigonometric structure. The obtained eigenvalues are expressed explicitly in terms of the hyperedge size, the number of directional families, and the lattice side length, thereby capturing how the geometry of higher-order interactions shapes the spectral structure.