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

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,S-m,S-1) whose underlying graph G(S-(n,S-m,S-1)) is obtained from the underlying graph G(S-(n,S-m)) of the (n+m)-by-(n+m) double star sign patterns S-(n,S-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,S-m,S-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,S-m,S-1).