Reduced graph divisors

Let $G$ be a graph and $v_0 \in V(G)$. A divisor $D$ of $G$ is called $v_0$-reduced if:

1. $f (v) \geq 0$ for all $v ∈ V(G)- v_0$;
2. and for every non-empty set $S \subseteq V (G) − v_0$, there exists a vertex $v \in S$ such that $f (v) \lt d^-_S (v)$ (where $d^-_S (v)$ denotes the number of edges of $v$ leaving $S$).

Remarks

• The definition makes sense for directed graphs but is only interesting in the case of undirected (multi)graphs.
• The following theorem plays a fundamental role in graph divisor theory:
Theorem[1] Let $G$ be an undirected graph and let $v_0 \in V(G)$ be a fixed base vertex. Then for every divisor $D$ of $G$, there exists a unique $v_0$-reduced divisor $D$ such that $D \sim D'$ (here $\sim$ denotes the linear equivalence of graph divisors).
• As a consequence, elements of the Picard group of a graph can be represented by reduced divisors.

References

1. M. Baker, S. Norine, Riemann--Roch and Abel--Jacobi theory on a finite graph, Advances in Mathematics (2007) DOI link, ArXiv Link