DIMACS - Graduate Student Combinatorics Seminar

Title: Jacobian Groups of Graphs

Speaker: Louis Gaudet, Rutgers University

Date: Wednesday, April 4, 2018 12:15pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ


Given a (finite) graph G, there is a natural finite abelian group we can associate to G, called its Jacobian group. There are different sources of motivation for studying these groups. They are the "tropical" (i.e. piecewise-linear) analogs of classical algebro-geometric objects. They are used as models for certain physical systems so-called "sandpile groups." The statistics of Jacobian groups of random graphs are related to the Cohen-Lenstra heuristics, which describe statistics of ideal class groups of number fields. For instance, a fun fact: the Jacobian of a random graph is cyclic just about 79% of the time.

Rather than discuss these connections in depth, we´ll explore a more basic question: which groups actually appear as the Jacobian of some graph? Along the way, we´ll observe and exploit connections of the Jacobian group to classical graph theoretic properties of interest: planarity, counts of spanning trees, etc. We´ll construct graphs whose Jacobians realize a large class of groups, and we´ll prove that there are infinite families of groups that cannot be realized as the Jacobian of any graph. The general question, however, is still open, and I will point out several possible research problems.

See: http://sites.math.rutgers.edu/~yb165/GCS.html