Potential eventual positivity of sign patterns with the underlying broom graph

A sign pattern is a matrix whose entries belong to the set {+,−,0}. An n-by-nsign pattern A is said to allow an eventually positive matrix or be potentially eventually positive if there exist at least one real matrix A with the same sign pattern as A and a positive integer k0 such that Ak>0 for all k≥k0. Identifying the necessary and sufficient conditions for an n-by-n sign pattern to be potentially eventually positive, and classifying the n-by-n sign patterns that allow an eventually positive matrix were posed as two open problems by Berman, Catral, Dealba, et al. In this article, we focus on the potential eventual positivity of a collection of the n-by-n tree sign patterns An,4 whose underlying graph G(An,4) consists of a path P with 4 vertices, together with (n−4) pendent vertices all adjacent to the same end vertex of P. Some necessary conditions for the n-by-n tree sign patterns An,4 to be potentially eventually positive are established. All the minimal subpatterns of An,4 that allow an eventually positive matrix are identified. Consequently, all the potentially eventually positive subpatterns of An,4 are classified