Interlacing Results for Hypergraphs

Abstract: Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of complex networks with higher-order interactions, and their spectral theory studies the qualitative properties that can be inferred from the spectrum, i.e. the multiset of the eigenvalues, of an operator associated to a hypergraph. It is expected that a small perturbation of a hypergraph, such as the removal of a few vertices or edges, does not lead to a major change of the eigenvalues. In particular, it is expected that the eigenvalues of the original hypergraph interlace the eigenvalues of the perturbed hypergraph. Here we work on hypergraphs where, in addition, each vertex--edge incidence is given a real number, and we prove interlacing results for the adjacency matrix, the Kirchoff Laplacian and the normalized Laplacian. Tightness of the inequalities is also shown.