We determine the maximum order of an element in the critical group of a strongly
regular graph, and show that it achieves the spectral bound due to Lorenzini. We
extend the result to all graphs with exactly two nonzero Laplacian eigenvalues, and
study the signed graph version of the problem. We also study the monodromy pairing
on the critical groups, and suggest an approach to study the structure of these groups
using the pairing.
