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. We conjecture a relation between the group structure and the Laplacian spectrum for a large class of graphs having integer spectra (the decomposable graphs of Kelmans). Based on computer evidence, we conjecture the exact group structure for the well-studied subclass of threshold graphs, and prove this conjecture for the subclass which we call generic threshold graphs.