Vertex-transitive nut graph order-degree existence problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914252938280960 |
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| author | Damnjanović, Ivan |
| author_facet | Damnjanović, Ivan |
| contents | A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy $4 \mid d$, $d \ge 4$, $2 \mid n$ and $n \ge d + 4$; or $d \equiv 2 \pmod 4$, $d \ge 6$, $4 \mid n$ and $n \ge d + 6$. Here, we prove that for each such $n$ and $d$, there exists a $d$-regular Cayley nut graph of order $n$. As a direct consequence, we obtain all the pairs $(n, d)$ for which there is a $d$-regular vertex-transitive (resp. Cayley) nut graph of order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex-transitive nut graph order-degree existence problem Damnjanović, Ivan Combinatorics Number Theory 05C50, 05C25, 11C08, 12D05 A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy $4 \mid d$, $d \ge 4$, $2 \mid n$ and $n \ge d + 4$; or $d \equiv 2 \pmod 4$, $d \ge 6$, $4 \mid n$ and $n \ge d + 6$. Here, we prove that for each such $n$ and $d$, there exists a $d$-regular Cayley nut graph of order $n$. As a direct consequence, we obtain all the pairs $(n, d)$ for which there is a $d$-regular vertex-transitive (resp. Cayley) nut graph of order $n$. |
| title | Vertex-transitive nut graph order-degree existence problem |
| topic | Combinatorics Number Theory 05C50, 05C25, 11C08, 12D05 |
| url | https://arxiv.org/abs/2507.02481 |