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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.17737 |
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| _version_ | 1866912949440872448 |
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| author | yaning, Jia Pan, Shengyong |
| author_facet | yaning, Jia Pan, Shengyong |
| contents | In this paper, we give an explicit formula for the rank of the $Q$-walk matrix of the Dynkin graph $A_n$. Moreover, we prove that its Smith normal form is $$ \mathrm{diag}\left( \underset{r=\lceil \frac{n}{2} \rceil}{\underbrace{1,2,2,...,2}},0,...,0 \right), $$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left( A_n \right) $ of the Dynkin graph $A_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Smith normal form of the Q-walk matrix of the Dynkin graph $A_n$ yaning, Jia Pan, Shengyong Rings and Algebras Combinatorics In this paper, we give an explicit formula for the rank of the $Q$-walk matrix of the Dynkin graph $A_n$. Moreover, we prove that its Smith normal form is $$ \mathrm{diag}\left( \underset{r=\lceil \frac{n}{2} \rceil}{\underbrace{1,2,2,...,2}},0,...,0 \right), $$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left( A_n \right) $ of the Dynkin graph $A_n$. |
| title | The Smith normal form of the Q-walk matrix of the Dynkin graph $A_n$ |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2411.17737 |