A bound on the scrambling index of a primitive matrix using Boolean rank

Abstract

The scrambling index of an n × n primitive matrix A is the smallest positive integer k such that A k ( A t ) k = J , where A t denotes the transpose of A and J denotes the n × n all ones matrix. For an m × n Boolean matrix M , its Boolean rank b ( M ) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b × n Boolean matrix B . In this paper, we give an upper bound on the scrambling index of an n × n primitive matrix M in terms of its Boolean rank b ( M ) . Furthermore we characterize all primitive matrices that achieve the upper bound.