Chip-firing based methods in the Riemann--Roch theory of directed graphs

TL;DR

This paper extends the Riemann--Roch theorem to directed graphs using chip-firing methods, providing new proofs and exploring the limits of Riemann--Roch equalities in various classes of directed graphs.

Contribution

It introduces a chip-firing based approach to Riemann--Roch theory on directed graphs, including a simplified proof for Eulerian graphs and analysis of Riemann--Roch equalities in strongly connected digraphs.

Findings

Proved Riemann--Roch inequality for Eulerian directed graphs.

Identified limitations of Riemann--Roch equalities in strongly connected digraphs.

Provided examples illustrating the scope of Riemann--Roch-type results.

Abstract

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 of Bj\"orner, Lov\'asz and Shor. We use this connection to prove Riemann--Roch-type results on directed graphs. We give a simple proof for a Riemann--Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann--Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.