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arXiv · 2508.05761

The gonality of circulant graphs

Lisa Cenek·Lizzie Ferguson·Eyobel Gebre·Cassandra Marcussen·Jason Meintjes·Ralph Morrison·Liz Ostermeyer·Shefali Ramakrishna
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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 \(10\).

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Lisa Cenek, Lizzie Ferguson, Eyobel Gebre, Cassandra Marcussen, Jason Meintjes, Ralph Morrison, Liz Ostermeyer, Shefali Ramakrishna. 2025-08-07. The gonality of circulant graphs. https://arxiv.org/abs/2508.05761

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