Interlacing inequalities for eigenvalues of discrete Laplace operators

TL;DR

This paper develops topological methods to establish interlacing inequalities for eigenvalues of discrete Laplace operators on weighted simplicial complexes, generalizing classical graph theory results and providing new eigenvalue bounds.

Contribution

It introduces a systematic topological approach to derive eigenvalue inequalities for Laplacians on simplicial complexes, extending known graph results to higher dimensions.

Findings

Derived eigenvalue bounds for Laplacians on simplicial complexes.

Unified framework generalizing graph interlacing inequalities.

Controlled spectral effects of complex operations like deletion and contraction.

Abstract

The term interlacing refers to systematic inequalities between the sequences of eigenvalues of two operators defined on objects related by a specific oper- ation. In particular, knowledge of the spectrum of one of the objects then implies eigenvalue bounds for the other one. In this paper, we therefore develop topological arguments in order to de- rive such analytical inequalities. We investigate, in a general and systematic manner, interlacing of spectra for weighted simplicial complexes with arbi- trary weights. This enables us to control the spectral effects of operations like deletion of a subcomplex, collapsing and contraction of a simplex, cover- ings and simplicial maps, for absolute and relative Laplacians. It turns out that many well-known results from graph theory become special cases of our general results and consequently admit improvements and generalizations. In particular, we derive a number of effective eigenvalue bounds.