Possible isolation number of a matrix over nonnegative integers

Beasley, LeRoy B., Jun, Young Bae, and Song, Seok-Zun. "Possible isolation number of a matrix over nonnegative integers." Czechoslovak Mathematical Journal 68.4 (2018): 1055-1066. <http://eudml.org/doc/294383>.

@article{Beasley2018,
abstract = {Let $\mathbb \{Z\}_+$ be the semiring of all nonnegative integers and $A$ an $m\times n$ matrix over $\mathbb \{Z\}_+$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m\times k$ matrix times a $k\times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2\times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the rank of $A$ and the Boolean rank of the support of $A$. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is $1$ or $2$ only. We also determine a special type of $m \times n$ matrices whose isolation number is $m$. That is, those matrices are permutationally equivalent to a matrix $A$ whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.},
author = {Beasley, LeRoy B., Jun, Young Bae, Song, Seok-Zun},
journal = {Czechoslovak Mathematical Journal},
keywords = {rank; Boolean rank; isolated entry; isolation number},
language = {eng},
number = {4},
pages = {1055-1066},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Possible isolation number of a matrix over nonnegative integers},
url = {http://eudml.org/doc/294383},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Beasley, LeRoy B.
AU - Jun, Young Bae
AU - Song, Seok-Zun
TI - Possible isolation number of a matrix over nonnegative integers
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 1055
EP - 1066
AB - Let $\mathbb {Z}_+$ be the semiring of all nonnegative integers and $A$ an $m\times n$ matrix over $\mathbb {Z}_+$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m\times k$ matrix times a $k\times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2\times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the rank of $A$ and the Boolean rank of the support of $A$. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is $1$ or $2$ only. We also determine a special type of $m \times n$ matrices whose isolation number is $m$. That is, those matrices are permutationally equivalent to a matrix $A$ whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.
LA - eng
KW - rank; Boolean rank; isolated entry; isolation number
UR - http://eudml.org/doc/294383
ER -