Interlacing inequalities for eigenvalues of discrete Laplace operators

Abstract: The term interlacing refers to systematic inequalities between the sequences of eigenvalues of two operators defined on objects related by a specific oper- ation. 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 de- rive such analytical inequalities. We investigate, in a general and systematic manner, interlacing of spectra for weighted simplicial complexes with arbi- trary weights. This enables us to control the spectral effects of operations like deletion of a subcomplex, collapsing and contraction of a simplex, cover- ings 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.