We present briefly the theory of divisors on graphs, parallel to that  ofline bundles on curves. We show some tools for computing the ranks of spin structures. In general the rank singles out a distinguished noneffective spin structure, but the parity of the rank is not a quadratic form. In the case of hyperelliptic spin structures the parity is a Z/2-quadratic form. We take this as an opportunity to detail the interplay between combinatorial and algebraic rank over the moduli space.
This is work in collaboration with Marco Pacini.
org: Turchet Amos