Algebraic Connectivity of Finite-element Hypergraphs

This paper generalizes results that relate the connectivity of a weighted graph to the smallest nonzero eigenvalue of the graph’s Laplacian matrix. We generalize these results to hypergraphs with vector-valued vertices and matrix-valued edges. Our definitions of hypergraphs and their connectivity are designed to model finite-element meshes. The physical interpretation of our results is as follows. We say that a structure modeled by a finiteelement model is weak if its vertices can be displaced by a unit perturbation orthogonal to its rigid motions with only a small investment of energy (or work, virtual work, etc.) Given a partition of the vertices of a finiteelement model, a displacement is a cut displacement if under that displacement, energy is expanded only in elements incident on vertices in both subsets of the partition. Our goal is to show that a structure is weak if and only if it has a weak cut. Our analysis is purely algebraic and can be applied to finite-elements models from several application domains. Our ultimate objective is to find ways to construct socalled support preconditioners ([2],[3]) for finite-element matrices. The analysis of existing support preconditioners is based on relating generalized eigenvalues of the Laplacians to complex combinatorial properties of pairs of their graphs. We believe that studying the simpler relationship of a simple eigenvalue to the connectivity of a hypergraph will eventually help us develop support preconditioners for hypergraphs.