Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2405.01863 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916425001598976 |
|---|---|
| author | Gregor, Petr Hoang, Hung P. Merino, Arturo Mička, Ondřej |
| author_facet | Gregor, Petr Hoang, Hung P. Merino, Arturo Mička, Ondřej |
| contents | We show that all invertible $n \times n$ matrices over any finite field $\mathbb{F}_q$ can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or diving a row by the generator of the multiplicative group of $\mathbb{F}_q$. This even holds if the addition and subtraction of each row is allowed to some specific rows satisfying a certain mild condition. Moreover, we can prescribe the first and the last matrix if $n\ge 3$, or $n=2$ and $q>2$. In other words, the corresponding flip graph on all invertible $n \times n$ matrices over $\mathbb{F}_q$ is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generating all invertible matrices by row operations Gregor, Petr Hoang, Hung P. Merino, Arturo Mička, Ondřej Combinatorics We show that all invertible $n \times n$ matrices over any finite field $\mathbb{F}_q$ can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or diving a row by the generator of the multiplicative group of $\mathbb{F}_q$. This even holds if the addition and subtraction of each row is allowed to some specific rows satisfying a certain mild condition. Moreover, we can prescribe the first and the last matrix if $n\ge 3$, or $n=2$ and $q>2$. In other words, the corresponding flip graph on all invertible $n \times n$ matrices over $\mathbb{F}_q$ is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs. |
| title | Generating all invertible matrices by row operations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.01863 |