rank; Boolean rank; isolated entry; isolation number
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.
 Bondy, J. A., Murty, U. S. R.: Graph Theory
. Graduate texts in Mathematics 244, Springer, Berlin (2008). MR 2368647
| Zbl 1134.05001
 Caen, D. de, Gregory, D. A., Pullman, N. J.: The Boolean rank of zero-one matrices
. Combinatorics and Computing Proc. 3rd Caribb. Conf., Cave Hill/Barbados (1981), 169-173. MR 0657202
| Zbl 0496.20052