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Annals of Global Analysis and Geometry

, Volume 43, Issue 2, pp 177–207 | Cite as

Interlacing inequalities for eigenvalues of discrete Laplace operators

  • Danijela HorakEmail author
  • Jürgen Jost
Article

Abstract

The term “interlacing” refers to systematic inequalities between the sequences of eigenvalues of two operators defined on objects related by a specific operation. 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 derive such analytical inequalities. We investigate, in a general and systematic manner, interlacing of spectra for weighted simplicial complexes with arbitrary weights. This enables us to control the spectral effects of operations like deletion of a subcomplex, collapsing and contraction of a simplex, coverings 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.

Keywords

Weight Function Laplace Operator Simplicial Complex Weighted Graph Laplacian Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of Mathematics and Computer ScienceLeipzig UniversityLeipzigGermany
  3. 3.Santa Fe Institute for the Sciences of ComplexitySanta FeUSA

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