Czechoslovak Mathematical Journal
Springer Berlin Heidelberg
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.
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.