We can associate a "dual graph" to each nodal curve. Analogously to the theory of divisors on curves, we can define the divisor group and Picard group of a graph. These two theories are then related by the specialization map: a map from Pic(X) to Pic(G), where G is the dual graph of the special fiber of a certain model of X. We then give an example of how the combinatorics of the graph can be used to determine information about the original curve by computing the gonality of a generic bidegree (m,n) curve on P1 x P1.