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 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.