  Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a
simplex $\sigma\notin X$ such that $\tau\in X$ for any $\tau\subsetneq \sigma$.
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.
