The Minimal Solutions of Boolean Matrix-equation A~k=J

摘要

Let A be a primitive Boolean matrix. γ(A) is the least number k such that A~k=J. σ(A) is the number of 1-entries in A. In this paper, the parameter N'(k,n)=min{σ(A) A~T =A, trace(A)=0, γ(A)k} is considered. Furthermore, we describe the set EG(k,n)= G(A)σ(A)=N'(k,n), A~t=A, trace (A)=0, γ(A)=k} and obtain a characterization of The minimal solutions with zero trace of the Boolean matrix equation A~k=J.