Spectral gaps, missing faces and minimal degrees

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $sigma otin X$ such that $tauin X$ for any $tausubsetneq sigma$. For a $k$-dimensional simplex $sigma$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $delta_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $mu_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $mu_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: [ mu_k(X)geq (d+1)(delta_k+k+1)-d n. ] 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.