  We analyze a new spectral graph matching algorithm, GRAph Matching by
Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex
correspondence between two unlabeled, edge-correlated weighted graphs.
Extending the exact recovery guarantees established in the companion paper for
Gaussian weights, in this work, we prove the universality of these guarantees
for a general correlated Wigner model. In particular, for two Erd\H{o}s-R\'enyi
graphs with edge correlation coefficient $1-\sigma^2$ and average degree at
least $\operatorname{polylog}(n)$, we show that GRAMPA exactly recovers the
latent vertex correspondence with high probability when $\sigma \lesssim
1/\operatorname{polylog}(n)$. Moreover, we establish a similar guarantee for a
variant of GRAMPA, corresponding to a tighter quadratic programming relaxation
of the quadratic assignment problem. Our analysis exploits a resolvent
representation of the GRAMPA similarity matrix and local laws for the
resolvents of sparse Wigner matrices.
