Combinatorial Laplace operators: a unifying approach, normalization and spectra

Danijela Horak

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Submission date: 08. Jun. 2011
Pages: 29
Bibtex
MSC-Numbers: 15A18, 05E45, 05C50
Keywords and phrases: combinatorial Laplacian, spectra, simplicial complex
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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 Δ_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.