Spectral gaps, missing faces and minimal degrees

Abstract

Let X be a simplicial complex with n vertices. A missing face of X is a simplex σ∉X such that τ∈X for any τ⊊σ. For a k-dimensional simplex σ in X, its degree in X is the number of (k+1)-dimensional simplices in X containing it. Let δk denote the minimal degree of a k-dimensional simplex in X. Let Lk denote the k-Laplacian acting on real k-cochains of X and let μk(X) denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps μk(X), for complexes X without missing faces of dimension larger than d:μk(X)≥(d+1)(δk+k+1)−dn. As a consequence we obtain a new proof of a vanishing result for the homology of simplicial complexes without large missing faces. We present a family of examples achieving equality at all dimensions, showing that the bound is tight. For d=1 we characterize the equality case.