Algebraic and combinatorial rank of divisors on finite graphs
  We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general. 
Lucia Caporaso, Yoav Len, and Margarida Melo
math.AG math.CO