The gonality of circulant graphs

TL;DR

This paper investigates the gonality of circulant graphs, providing universal bounds and exact values for specific classes like the Harary and antiprism graphs, advancing understanding of chip-firing complexities on these symmetric structures.

Contribution

It introduces a universal upper bound on the gonality of circulant graphs and computes exact gonality values for certain classes such as the Harary and antiprism graphs.

Findings

Universal upper bound on gonality for circulant graphs.

Gonality of the 4-regular Harary graph is 10 for large n.

Sufficiently large antiprism graphs have gonality 10.

Abstract

The gonality of a graph measures how difficult it is to move chips around the entirety of a graph according to certain chip-firing rules without introducing debt. In this paper we study the gonality of circulant graphs, a class of vertex-transitive graphs that can be specified by their number of vertices together with a list of cyclic adjacency relations satisfied by all vertices. We provide a universal upper bound on the gonality of all circulant graphs with a fixed adjacency list, which holds irrespective of the number of vertices. We use this upper bound together with computational methods to determine that the gonality of the \(4\)-regular Harary graph on \(n\) vertices is \(10\) for \(n\geq 16\). As a special case, this gives the gonality of sufficiently large antiprism graphs to be \(10\).