Critical groups of strongly regular graphs and their generalizations

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.