Exponents of primitive symmetric companion matrices

TL;DR

This paper characterizes the exponents of primitive symmetric companion matrices, providing formulas and counts for primitive and imprimitive cases, and fully describing the exponent set for this matrix class.

Contribution

It introduces formulas to compute exponents of primitive symmetric companion matrices and counts their occurrences, advancing understanding of their combinatorial properties.

Findings

Derived formulas for exponents of primitive matrices

Counted primitive and imprimitive symmetric companion matrices

Fully characterized the exponent set for primitive matrices

Abstract

A {\it symmetric companion matrix} is a matrix of the form $A +A^T$ where $A$ is a companion matrix all of whose entries are in $\{0,1\}$ and $A^T$ is the transpose of $A.$ In this paper, we find the total number of primitive and the total number of imprimitive symmetric companion matrices. We establish formulas to compute the exponent of every primitive symmetric companion matrix. Hence the exponent set for the class of primitive symmetric companion matrices is completely characterized. We also obtain the number of primitive symmetric companion matrices with a given exponent for certain cases.