Matrix periods and competition periods of Boolean Toeplitz matrices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913549071155200 |
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| author | Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Ryu, Homoon |
| author_facet | Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Ryu, Homoon |
| contents | In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13652 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Matrix periods and competition periods of Boolean Toeplitz matrices Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Ryu, Homoon Combinatorics 05C20, 05C50, 15B05 In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view. |
| title | Matrix periods and competition periods of Boolean Toeplitz matrices |
| topic | Combinatorics 05C20, 05C50, 15B05 |
| url | https://arxiv.org/abs/2208.13652 |