Jacobians of Nearly Complete Graphs

Graph Theory Seminar
Thursday, October 22, 2009 - 12:05pm for 1.5 hours (actually 80 minutes)
Skiles 255
Peter Whalen – Math, GT
Robin Thomas
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 talk I will present a combinatorial method of approaching the Jacobian of graphs by way of a chip-firing game played on its vertices. We then apply this chip-firing game to explicitly characterize the Jacobian of nearly complete graphs, those obtained from the complete graph by deleting either a cycle or two vertex-disjoint paths incident with all but one vertex. This is joint work with Sergey Norin.