  We study the spectral implications of re-weighting a graph by the
$\ell_\infty$-Lewis weights of its edges. Our main motivation is the
ER-Minimization problem (Saberi et al., SIAM'08): Given an undirected graph
$G$, the goal is to find positive normalized edge-weights $w\in \mathbb{R}_+^m$
which minimize the sum of pairwise \emph{effective-resistances} of $G_w$
(Kirchhoff's index). By contrast, $\ell_\infty$-Lewis weights minimize the
\emph{maximum} effective-resistance of \emph{edges}, but are much cheaper to
approximate, especially for Laplacians. With this algorithmic motivation, we
study the ER-approximation ratio obtained by Lewis weights.
  Our first main result is that $\ell_\infty$-Lewis weights provide a constant
($\approx 3.12$) approximation for ER-minimization on \emph{trees}. The proof
introduces a new technique, a local polarization process for
effective-resistances ($\ell_2$-congestion) on trees, which is of independent
interest in electrical network analysis. For general graphs, we prove an upper
bound $\alpha(G)$ on the approximation ratio obtained by Lewis weights, which
is always $\leq \min\{ \text{diam}(G), \kappa(L_{w_\infty})\}$, where $\kappa$
is the condition number of the weighted Laplacian. All our approximation
algorithms run in \emph{input-sparsity} time $\tilde{O}(m)$, a major
improvement over Saberi et al.'s $O(m^{3.5})$ SDP for exact ER-minimization.
  Finally, we demonstrate the favorable effects of $\ell_\infty$-LW reweighting
on the \emph{spectral-gap} of graphs and on their \emph{spectral-thinness}
(Anari and Gharan, 2015). En-route to our results, we prove a weighted analogue
of Mohar's classical bound on $\lambda_2(G)$, and provide a new
characterization of leverage-scores of a matrix, as the gradient (w.r.t
weights) of the volume of the enclosing ellipsoid.
