Interlacing inequalities for eigenvalues of discrete Laplace operators

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.