Jacobians of nearly complete and threshold graphs

The Jacobian of a graph, also known as the Picard group, sandpile group, or critical group, is a discrete analogue of the Jacobian of an algebraic curve. It is known that the order of the Jacobian of a graph is equal to its number of spanning trees, but the exact structure is known for only a few classes of graphs. In this paper, we compute the Jacobian for graphs of the form Kn∖E(H) where H is a subgraph of Kn on n−1 vertices that is either a cycle, or a union of two disjoint paths. We also offer a combinatorial proof of a result of Christianson and Reiner that describes the Jacobian for a subclass of threshold graphs.