Eventual Positivity of a Class of Double Star-like Sign Patterns

Ber-Lin Yu,Gu-Fang Mou,Zhongshan Li,Sanzhang Xu

doi:10.3390/sym14030512

Abstract

Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1).

Highlights

Received: 5 February 2022The study of combinatorial and qualitative information, which is only related to the signs of its entries and independent of the magnitudes of its entries of a real matrix, has attracted great attention
We investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G (S(n,m,1) ) is obtained from the underlying graph G (S(n,m) ) of the (n + m)-by-(n + m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices, and removing the edge between these center vertices
As an application of our results, we identify all the minimal potentially eventually exponentially positive sign pattern (MPEEP) sign patterns and classify all the potentially eventually exponentially positive (PEEP) sign patterns in the class of double star-like patterns S(n,m,1)

Summary

Introduction

Received: 5 February 2022The study of combinatorial and qualitative information, which is only related to the signs of its entries and independent of the magnitudes of its entries of a real matrix, has attracted great attention. Let S(n,m,1) = (αi,j ) be a symmetric double star-like sign pattern of order n + m + 1 of the form (∗) with α1,2 = α2,1 = − and α1,n+2 = αn+2,1 = +. Recall that every double star-like sign pattern S(n,m,1) that is PEP must be symmetric by Theorem 1.

Results

Conclusion