The set of stable indices of 0-1 matrices with a given order

TL;DR

This paper characterizes the set of stable indices for 0-1 matrices of a fixed size, providing a comprehensive understanding of their possible stability behaviors.

Contribution

It offers a complete characterization of the stable indices for 0-1 matrices of a given order, a novel classification in matrix theory.

Findings

Identifies all possible stable indices for 0-1 matrices of a fixed size.

Provides conditions under which certain stable indices occur.

Establishes a framework for analyzing stability in binary matrices.

Abstract

The stable index of a 0-1 matrix $A$ is defined to be the smallest integer $k$ such that $A^{k+1}$ is not a 0-1 matrix if such an integer exists; otherwise the stable index of $A$ is defined to be infinity. We characterize the set of stable indices of 0-1 matrices with a given order.