The stable index of 0-1 matrices

TL;DR

This paper introduces the stable index for 0-1 matrices, characterizes its maximum finite value for matrices of a given size, and identifies matrices that attain this maximum.

Contribution

It defines the stable index for 0-1 matrices and determines the maximum finite stable index for matrices of any order, including the matrices that achieve this maximum.

Findings

Maximum finite stable index of 0-1 matrices of order n determined.

Characterization of matrices attaining maximum stable index provided.

Stable index concept extends understanding of matrix powers in combinatorial contexts.

Abstract

We introduce the concept of stable index for 0-1 matrices. Let $A$ be a 0-1 square matrix. If $A^k$ is a 0-1 matrix for every positive integer $k$, then the stable index of $A$ is defined to be infinity; otherwise, the stable index of $A$ is defined to be the smallest positive integer $k$ such that $A^{k+1}$ is not a 0-1 matrix. We determine the maximum finite stable index of all 0-1 matrices of order $n$ as well as the matrices attaining the maximum finite stable index.