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

TL;DR

This paper establishes an upper bound on the scrambling index of primitive matrices based on their Boolean rank and characterizes matrices that attain this bound.

Contribution

It introduces a new bound relating the scrambling index to Boolean rank and characterizes matrices that reach this bound.

Findings

Upper bound on scrambling index in terms of Boolean rank

Characterization of matrices achieving the bound

Insight into the structure of primitive matrices

Abstract

The scrambling index of an $n\times n$ 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 $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its {\it Boolean rank} $b(M)$ is the smallest positive integer $b$ such that $M=AB$ for some $m \times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In this paper, we give an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b(M)$. Furthermore we characterize all primitive matrices that achieve the upper bound.