Spectra of combinatorial Laplace operators on simplicial complexes

Understand

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.