On $k$-idempotent 0-1 matrices

Let $k\ge 2$ be an integer. If a square 0-1 matrix $A$ satisfies $A^k=A$, then $A$ is said to be $k$-idempotent. In this paper, we give a characterization of $k$-idempotent 0-1 matrices. We also determine the maximum number of nonzero entries in $k$-idempotent 0-1 matrices of a given order as well as the $k$-idempotent 0-1 matrices attaining this maximum number.