  Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and
vertex conductance $\Psi^*(G)$. We show that there exists a symmetric,
stochastic matrix $P$, with off-diagonal entries supported on $E$, whose
spectral gap $\gamma^*(P)$ satisfies \[\Psi^*(G)^{2}/\log\Delta \lesssim
\gamma^*(P) \lesssim \Psi^*(G).\] Our bound is optimal under the Small Set
Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who
obtained such a result with $\log\Delta$ replaced by $\log|V|$.
  In order to obtain our result, we show how to embed a negative-type
semi-metric $d$ defined on $V$ into a negative-type semi-metric $d'$ supported
in $\mathbb{R}^{O(\log\Delta)}$, such that the (fractional) matching number of
the weighted graph $(V,E,d)$ is approximately equal to that of $(V,E,d')$.
