On cubic polycirculant nut graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910990813102080 |
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| author | Bašić, Nino Damnjanović, Ivan |
| author_facet | Bašić, Nino Damnjanović, Ivan |
| contents | A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms having $\ell$ vertex orbits of equal size. It is not difficult to observe that there exists no cubic $1$-circulant nut graph or cubic $2$-circulant nut graph, while the full classification of all the cubic $3$-circulant nut graphs was recently obtained [Electron. J. Comb. 31(2) (2024), #2.31]. Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \ge 4$ and show that there is no cubic $4$-circulant nut graph or cubic $5$-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $\ell$-circulant nut graphs for any fixed $\ell \in \{6, 7 \}$ or $\ell \ge 9$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On cubic polycirculant nut graphs Bašić, Nino Damnjanović, Ivan Combinatorics 05C50, 05C25, 11C08 A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms having $\ell$ vertex orbits of equal size. It is not difficult to observe that there exists no cubic $1$-circulant nut graph or cubic $2$-circulant nut graph, while the full classification of all the cubic $3$-circulant nut graphs was recently obtained [Electron. J. Comb. 31(2) (2024), #2.31]. Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \ge 4$ and show that there is no cubic $4$-circulant nut graph or cubic $5$-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $\ell$-circulant nut graphs for any fixed $\ell \in \{6, 7 \}$ or $\ell \ge 9$. |
| title | On cubic polycirculant nut graphs |
| topic | Combinatorics 05C50, 05C25, 11C08 |
| url | https://arxiv.org/abs/2411.16904 |