Nut graphs with a prescribed number of vertex and edge orbits
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912842200907776 |
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| author | Bašić, Nino Damnjanović, Ivan |
| author_facet | Bašić, Nino Damnjanović, Ivan |
| contents | A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \ge 2$ and any $k \ge r + 1$, there exist infinitely many nut graphs with $r$ vertex orbits and $k$ edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \ge 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20201 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nut graphs with a prescribed number of vertex and edge orbits Bašić, Nino Damnjanović, Ivan Combinatorics 05C50, 05C25 A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \ge 2$ and any $k \ge r + 1$, there exist infinitely many nut graphs with $r$ vertex orbits and $k$ edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \ge 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits. |
| title | Nut graphs with a prescribed number of vertex and edge orbits |
| topic | Combinatorics 05C50, 05C25 |
| url | https://arxiv.org/abs/2502.20201 |