Spectra of combinatorial Laplace operators on simplicial complexes

TL;DR

This paper develops a comprehensive framework for Laplace operators on simplicial complexes, analyzing their spectra and how various combinatorial modifications affect these spectral properties.

Contribution

It introduces a unified framework for Laplace operators on simplicial complexes and systematically studies spectral effects of topological modifications.

Findings

Spectral properties encode combinatorial features of complexes.

Wedge sum, join, and motif duplication influence the spectrum.

Normalized Laplace operator $ riangle_{i}^{up}$ is systematically investigated.

Abstract

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator $\Delta_{i}^{up}$ on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator, and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum.