The primitive Boolean matrices with the second largest scrambling index by Boolean rank

Shao, Yanling ; Gao, Yubin

Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 269-283.

Résumé

The scrambling index of an $n\times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^k(A^{\rm T})^k=J$, where $A^{\rm T}$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its 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 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b(M)$, and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.

Détail

DOI : 10.1007/s10587-014-0099-4

Classification : 05C20, 05C50, 05C75, 15B35
Keywords: scrambling index; primitive matrix; Boolean rank