  We propose a nonparametric two-sample test statistic for low-rank,
conditionally independent edge random graphs whose edge probability matrices
have negative eigenvalues and arbitrarily close eigenvalues. Our proposed test
statistic involves using the maximum mean discrepancy applied to suitably
rotated rows of a graph embedding, where the rotation is estimated using
optimal transport. We show that our test statistic, appropriately scaled, is
consistent for sufficiently dense graphs, and we study its convergence under
different sparsity regimes. In addition, we provide empirical evidence
suggesting that our novel alignment procedure can perform better than the
na\"ive alignment in practice, where the na\"ive alignment assumes an eigengap.
