  We present an adaptation of the Kato--Temple inequality for bounding
perturbations of eigenvalues with applications to statistical inference for
random graphs, specifically hypothesis testing and change-point detection. We
obtain explicit high-probability bounds for the individual distances between
certain signal eigenvalues of a graph's adjacency matrix and the corresponding
eigenvalues of the model's edge probability matrix, even when the latter
eigenvalues have multiplicity. Our results extend more broadly to the
perturbation of singular values in the presence of quite general random matrix
noise.
