Exponents of the primitive Boolean matrices with fixed girth

TL;DR

This paper characterizes the primitive exponents of primitive Boolean matrices with fixed girth, providing a complete description for matrices of order at least 10 and certain girth values.

Contribution

It offers a complete characterization of primitive exponents for primitive Boolean matrices with fixed girth under specified conditions.

Findings

Primitive matrices with girth g have exponents in a specific interval.

Complete characterization of matrices with exponents in the given range.

Results apply to matrices of order n ≥ 10 and girth g > (n^2 - 4n)/(4(n-3)).

Abstract

The $girth$ of a primitive Boolean matrix is defined to be the $girth$ of its associated digraph. In this paper, among all primitive Boolean matrices of order $n$, the primitive exponents of those of girth $g$ are considered. For the primitive matrices of both order $n\geq 10$ and girth $g>\frac{n^{2}-4n}{4(n-3)}$, the matrices with primitive exponents in $[2n-2 +(g- 1)(n-3), n+g(n-2)]$ are completely characterized.