  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 non-zero
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.
