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

Title:

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

Creator:

Shao, Yanling and Gao, Yubin

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:fc32fe0f-f9c1-4d6c-8dd0-f7d695075f53
uuid:fc32fe0f-f9c1-4d6c-8dd0-f7d695075f53

Subject:

scrambling index, primitive matrix, and Boolean rank

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

269-283

Source:

Czechoslovak Mathematical Journal | 2014 Volume:64 | Number:1

Harvested from:

CDK

Metadata only:

false