  Cheeger's inequality shows that any undirected graph $G$ with minimum nonzero
normalized Laplacian eigenvalue $\lambda_G$ has a cut with conductance at most
$O(\sqrt{\lambda_G})$. Qualitatively, Cheeger's inequality says that if the
relaxation time of a graph is high, there is a cut that certifies this.
However, there is a gap in this relationship, as cuts can have conductance as
low as $\Theta(\lambda_G)$.
  To better approximate the relaxation time of a graph, we consider a more
general object. Instead of bounding the mixing time with cuts, we bound it with
cuts in graphs obtained by Schur complementing out vertices from the graph $G$.
Combinatorially, these Schur complements describe random walks in $G$
restricted to a subset of its vertices. As a result, all Schur complement cuts
have conductance at least $\Omega(\lambda_G)$. We show that unlike with cuts,
this inequality is tight up to a constant factor. Specifically, there is a
Schur complement cut with conductance at most $O(\lambda_G)$.
