Interlacing Results for Hypergraphs

TL;DR

This paper extends interlacing eigenvalue results to hypergraphs with weighted incidences, analyzing how small perturbations affect spectral properties of various associated matrices.

Contribution

It introduces interlacing theorems for the adjacency, Kirchhoff Laplacian, and normalized Laplacian matrices of weighted hypergraphs, including tightness results.

Findings

Eigenvalues of hypergraphs interlace after perturbations

Interlacing inequalities are proven for multiple hypergraph matrices

Tightness of the inequalities is established

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.