Matrix periods and competition periods of Boolean Toeplitz matrices

TL;DR

This paper investigates the periodic properties of Boolean Toeplitz matrices, providing explicit formulas for their matrix and competition periods, and explores their structural limits using graph theoretic methods.

Contribution

It introduces new formulas for the matrix and competition periods of Boolean Toeplitz matrices based on subset parameters, and analyzes their asymptotic behavior through graph theory.

Findings

Matrix period is given by d/d' under certain conditions.

Competition period is always 1 under specified conditions.

The limit of the matrix sequence is a directed sum of all-ones matrices with zero diagonals.

Abstract

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.