Spectral gaps, missing faces and minimal degrees

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $\sigma \mathrm{\notin }X$ such that $\tau \in X$ for any $\tau ⊊\sigma$. 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 $d$: $${\mu }_k(X)\ge (d+1)({\delta }_k+k+1)-dn\mathrm{.}$$ 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.