Riemann--Roch theorem on directed graphs

Bálint Hujter, Lilla Tóthmérész

Published 2015-11-11Version 1

Baker and Norine proved a Riemann--Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game. Based on this connection, we give a new proof for the Riemann--Roch theorem on graphs which can be generalized to Eulerian directed graphs, improving a result of Amini and Manjunath. We also give a graph-theoretic version of the abstract Riemann--Roch criterion of Baker and Norine, and explore the natural Riemann--Roch property introduced by Asadi and Backman.