#### Document Type

Article

#### Journal/Book Title/Conference

Czechoslovak Mathematical Journal

#### Publisher

Springer Berlin Heidelberg

#### Publication Date

5-8-2018

#### First Page

1

#### Last Page

12

#### Abstract

Let ℤ_{+} be the semiring of all nonnegative integers and *A* an *m* × *n* matrix over ℤ_{+}. The rank of *A *is the smallest *k* such that *A* can be factored as an *m* × *k* matrix times a *k×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 × 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×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.

#### Recommended Citation

Beasley, LeRoy B.; Jun, Young Bae; and Song, Seok-Zun, "Possible Isolation Number of a Matrix Over Nonnegative Integers" (2018). *Mathematics and Statistics Faculty Publications.* Paper 232.

https://digitalcommons.usu.edu/mathsci_facpub/232

## Comments

The original publication is available at: https://articles.math.cas.cz/10.21136/CMJ.2018.0068-17