# Reduced graph divisors

From Egres Open

Let [math]G[/math] be a graph and [math]v_0 \in V(G)[/math]. A divisor [math]D[/math] of [math]G[/math] is called **[math]v_0[/math]-reduced** if:

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

## 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 [math]G[/math] be an undirected graph and let [math]v_0 \in V(G)[/math] be a fixed base vertex. Then for every divisor [math]D[/math] of [math]G[/math], there exists a unique [math]v_0[/math]-reduced divisor [math]D[/math] such that [math]D \sim D'[/math] (here [math]\sim[/math] 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

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