# Rank of graph divisors

The rank of divisor $D$ is the smallest number $G$ such that by subtracting a divisor of degree $k+1$ from $D$, we can get a divisor which is not linearly equivalent to an effective divisor.

Formally, let $Div(G)$ denote the set of divisors on a graph $G$. A graph divisor $D$ is effective if $D(v) \ge 0$ for all $v \in V$.

Definition(The rank of a divisor [1])

rank(D) = min {$\deg(E) - 1$: $E \in Div(G)$, $E$ is effective, $D-E$ is not linearly equivalent to any effective divisor}.

### References

1. M. Baker, S. Norine, Riemann--Roch and Abel--Jacobi theory on a finite graph, DOI link, ArXiv Link
2. V. Kiss, L. Tóthmérész, Chip-firing games on Eulerian digraphs and the NP-hardness of computing the rank of a divisor on a graph, DOI link, EGRES Technical Report