Graphs can be viewed as (non-Archimedean or tropical) analogues of Riemann surfaces. For example, there are well-behaved (and useful) notions of divisors, Riemann-Roch, and Abel-Jacobi theory on graphs. I introduce the notion of semibreak divisors, which provide nice representatives for effective divisor classes on graphs. I then discuss a few applications about the generic behavior of effective divisor classes, analogous to some classical results on Riemann surfaces. Proofs in this tropical setting are more subtle.
This is joint work with Andreas Gross (Imperial College) and Lilla Tóthmérész (Cornell).