On $k$-idempotent 0-1 matrices

TL;DR

This paper characterizes $k$-idempotent 0-1 matrices, determines their maximum nonzero entries for a given size, and identifies matrices that achieve this maximum, advancing understanding of their structure.

Contribution

It provides a complete characterization of $k$-idempotent 0-1 matrices and finds the maximum number of nonzero entries they can have.

Findings

Characterization of $k$-idempotent 0-1 matrices

Maximum number of nonzero entries in such matrices

Identification of matrices attaining the maximum

Abstract

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.