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Riemann-Roch and Abel-Jacobi theory on a finite graph. (English) Zbl 1124.05049
Summary: It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.

##### MSC:
 05C38 Paths and cycles 14H55 Riemann surfaces; Weierstrass points; gap sequences 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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