Matrix periods and competition periods of Boolean Toeplitz matrices

In this paper, we study the matrix period and competition of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S T {1,…,n−1}, an n×n A=Tn〈S;T〉 is defined to have 1 as (i,j)-entry if only j−i∈S or i−j∈T. We show that max⁡S+min⁡T≤n min⁡S+max⁡T≤n, then A has d/d′ where d=gcd⁡(s+t|s∈S,t∈T) d′=gcd⁡(d,min⁡S). Moreover, it shown limit sequence {Am(AT)m}m=1∞ directed sum all ones except zero diagonal. many literatures see graph theoretic method can be used prove strong structural properties about matrices. Likewise, develop our work from point view.