Gonality

The gonality (also called divisorial gonality)  of a (finite) graph  is the minimum degree of a rank 1 divisor on that graph. It can be thought of as the minimum number of chips that can be placed on that graph such that a debt of 1 can be eliminated via "chip-firing moves" over all possible debt placements.

The gonality of a graph is one of several graph analogs of the gonality of an algebraic curve, which is the minimum degree of a rational map from the curve to the projective line (Echavarria 2021).

The gonality of a graph and satisfies

where  is the vertex connectivity,  is the edge connectivity,  is the treewidth, and  is the scramble number of  (Harp et al. 2020, Echavarria et al. 2021).

Trees have gonality of 1. Gonalities for a number of special classes of graphs are summarized by Echavarria et al. (2021, Examples 2.7 and 2.8).

The gonality of a graph is NP-hard to compute (Gijswijt et al. 2020, Echavarria 2021).

REFERENCES

Echavarria, M.; Everett, M.; Huang, R.; Jacoby, L.; Morrison, R.; Weber, B. "On the Scramble Number of Graphs." 29 Mar 2021. https://arxiv.org/abs/2103.15253.

Gijswijt, D.; Smit, H.; and van der Wegen, M. "Computing Graph Gonality Is Hard." Disc. Appl. Math. 287, 134-149, 2020.

Harp, M.; Jackson, E.; Jensen, D.; and Speeter, N. "A New Lower Bound on Graph Gonality." 1 Jun 2020. https://arxiv.org/abs/2006.01020.