The critical group of a threshold graph

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph. The structure of this group is a subtle isomorphism invariant that has received much attention recently, partly due to its relation to the graph Laplacian and chip-firing games. However, the group structure has been determined for relatively few classes of graphs. Based on computer evidence, we conjecture the exact group structure for a well-studied class of graphs having integer spectra, the threshold graphs, and prove this conjecture for the subclass which we call generic threshold graphs.