0-1 matrices with zero trace whose squares are 0-1 matrices

TL;DR

This paper investigates the maximum number of nonzero entries in 0-1 matrices with zero trace, whose squares are also 0-1 matrices, and characterizes the extremal matrices for large sizes.

Contribution

It determines the maximum nonzero entries and characterizes extremal matrices for 0-1 matrices with zero trace and specific square properties when size is at least 8.

Findings

Maximum number of nonzero entries identified

Extremal matrices characterized for large n

Results applicable to matrices with zero trace and square constraints

Abstract

In this paper, we determine the maximum number of nonzero entries in 0-1 matrices of order $n$ with zero trace whose squares are 0-1 matrices when $n\ge 8$. The extremal matrices attaining this maximum number are also characterized.