# Duality between chip-firing games and graph divisor theory

There is a duality between the notion of graph divisor theory and chip-firing, which was discovered by Baker and Norine [1].

Let $G$ be a graph and let $K^+ = K^+_G$ be the chip-distribution with $K^+(v)=d(v) - 1$ for each vertex $v$.

For a divisor $D\in Div(G)$ with $D(v)\leq d(v)-1$ for each $v\in V(G)$, we call $K^+ - D\geq 0$ the dual pair of $D$. Note that each chip-distribution is a dual pair of some divisor.

The following proposition of Baker and Norine is the key ingredient of the duality:

Theorem[1]. For a divisor $D\in Div(G)$ with $D(v)\leq d(v)-1$ for each $v\in V(G)$, there exists an effective divisor equivalent to $D$ if and only if $K^+ - D$ is a terminating chip-distribution.

The following is a straightforward consequence of the above theorem.

Theorem. Let $D \in Div(G)$ be a divisor with $f(v)\leq d(v)-1$ for each $v\in V(G)$, and let $x \in Chip(G)$ be its dual pair. Then $rank(f) = dist(x) - 1$.

### References

1. M. Baker, S. Norine, Riemann--Roch and Abel--Jacobi theory on a finite graph, DOI link, ArXiv Link