The gonality of circulant graphs
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918118472810496 |
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| author | Cenek, Lisa Ferguson, Lizzie Gebre, Eyobel Marcussen, Cassandra Meintjes, Jason Morrison, Ralph Ostermeyer, Liz Ramakrishna, Shefali |
| author_facet | Cenek, Lisa Ferguson, Lizzie Gebre, Eyobel Marcussen, Cassandra Meintjes, Jason Morrison, Ralph Ostermeyer, Liz Ramakrishna, Shefali |
| contents | 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 \(10\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The gonality of circulant graphs Cenek, Lisa Ferguson, Lizzie Gebre, Eyobel Marcussen, Cassandra Meintjes, Jason Morrison, Ralph Ostermeyer, Liz Ramakrishna, Shefali Combinatorics Algebraic Geometry 14T99, 05C57 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 \(10\). |
| title | The gonality of circulant graphs |
| topic | Combinatorics Algebraic Geometry 14T99, 05C57 |
| url | https://arxiv.org/abs/2508.05761 |