On Arithmetical Structures on Complete Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866914648105680896 |
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| author | Harris, Zachary Louwsma, Joel |
| author_facet | Harris, Zachary Louwsma, Joel |
| contents | An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor each of which divides their sum. We show that, for all positive integers $c$ less than a certain bound depending on $n$, there is an arithmetical structure on $K_n$ with largest value $c$. We also show that, if each prime factor of $c$ is greater than $(n+1)^2/4$, there is no arithmetical structure on $K_n$ with largest value $c$. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on $K_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_02022 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On Arithmetical Structures on Complete Graphs Harris, Zachary Louwsma, Joel Number Theory Combinatorics 11D68 (Primary) 05C50, 11A41 (Secondary) An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor each of which divides their sum. We show that, for all positive integers $c$ less than a certain bound depending on $n$, there is an arithmetical structure on $K_n$ with largest value $c$. We also show that, if each prime factor of $c$ is greater than $(n+1)^2/4$, there is no arithmetical structure on $K_n$ with largest value $c$. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on $K_n$. |
| title | On Arithmetical Structures on Complete Graphs |
| topic | Number Theory Combinatorics 11D68 (Primary) 05C50, 11A41 (Secondary) |
| url | https://arxiv.org/abs/1909.02022 |