Graph gonality gon(G): minimum degree of a divisor with positive rank in chip-firing theory. Place chips on vertices, fire vertices to redistribute. gon(tree)=1, gon(C_n)=2, gon(K_n)=n-1. Tropical analog of algebraic curve gonality.
How to Use
Select graph:
- gon: Gonality
- Chips: Minimum config
- Rank: Divisor rank
Chip-Firing
Place chips on vertices. 'Fire' a vertex: send one chip along each edge. A divisor D has rank ≥ r if after removing r chips from any vertex, D can reach an effective (all-nonneg) divisor via chip-firing moves.
Tropical Geometry
Graph gonality = tropical curve gonality. Baker-Norine theorem (2007): graphs satisfy Riemann-Roch! r(D) - r(K-D) = deg(D) - g + 1, where g = |E|-|V|+1. Beautiful algebra-geometry-combinatorics connection.
Step-by-Step Instructions
- 1Select graph.
- 2Compute gonality.
- 3Find optimal divisor.
- 4Apply chip-firing.
- 5Check Riemann-Roch.