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MiS Preprint
34/2011

Combinatorial Laplace operators: a unifying approach, normalization and spectra

Danijela Horak

Abstract

In this paper we present a general framework for the systematic study of all known types of combinatorial Laplace operators i.e. the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, the normalized graph Laplacian. Furthermore, we define normalized Laplace operator $\Delta_{i}^{up}$ on simplicial complexes and present its basic properties. The effects of a wedge sum, a join and a duplication of a motif on the spectrum of normalized Laplace operator are investigated, and some of the combinatorial features of a simplicial complex that are encoded in its spectrum are identified.

Received:
Jun 8, 2011
Published:
Jun 27, 2011
MSC Codes:
15A18, 05E45, 05C50
Keywords:
combinatorial Laplacian, spectra, simplicial complex

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Preprint
2011 Repository Open Access
Danijela Horak

Combinatorial Laplace operators : a unifying approach, normalization and spectra