Let G=(V,E) be an undirected graph with maximum degree Δ and
vertex conductance Ψ∗(G). We show that there exists a symmetric,
stochastic matrix P, with off-diagonal entries supported on E, whose
spectral gap γ∗(P) satisfies Ψ∗(G)2/logΔ≲γ∗(P)≲Ψ∗(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 logΔ 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 RO(logΔ), such that the (fractional) matching number of
the weighted graph (V,E,d) is approximately equal to that of (V,E,d′).Comment: 6 page