Spectra of combinatorial Laplace operators on simplicial complexes

Details Spectra of combinatorial Laplace operators on simplicial complexes Spectra of combinatorial Laplace operators on simplicial complexes

2011-05-13

arxiv.org/abs/1105.2712v3

Mathematics

Algebraic Topology

46 pages, 5 figures

Scientific paper

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.

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