Sign pattern matrices associated with cycle graphs that require algebraic positivity

TL;DR

This paper characterizes sign pattern matrices linked to cycle graphs that guarantee algebraic positivity, providing insights into the conditions under which all matrices with such patterns are algebraically positive.

Contribution

It offers a complete characterization of sign pattern matrices associated with cycle graphs that ensure algebraic positivity, a novel contribution to matrix theory.

Findings

Characterization of sign pattern matrices requiring algebraic positivity

Conditions under which all matrices with a given sign pattern are algebraically positive

Extension of algebraic positivity concepts to cycle graph-related matrices

Abstract

A real matrix is said to be positive if its every entry is positive, and a real square matrix A is algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. A sign pattern matrix A is said to require a property if all matrices having sign pattern as A have that property. In this paper, we characterize all sign pattern matrices associated with cycle graphs that require algebraic positivity.