Linear operators that preserve graphical properties of matrices: isolation numbers

Title:

Linear operators that preserve graphical properties of matrices: isolation numbers

Creator:

Beasley, LeRoy B., Song, Seok-Zun, and Jun, Yizng Bae

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:7138ae47-3829-4019-91dc-24f9d591835b
uuid:7138ae47-3829-4019-91dc-24f9d591835b

Subject:

Boolean matrix, Boolean rank, Boolean linear operator, and isolation number

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of the set $S$ into the complement of the set $S$. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that $T$ is a Boolean linear operator that strongly preserves isolation number $k$ for any $1\leq k\leq \min \{m,n\}$ if and only if there are fixed permutation matrices $P$ and $Q$ such that for $X\in {\mathcal M}_{m,n}(\mathbb B)$ $T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.

Language:

English

Rights:

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

Coverage:

819-826

Source:

Czechoslovak Mathematical Journal | 2014 Volume:64 | Number:3

Harvested from:

CDK

Metadata only:

false