A sign pattern is a matrix whose entries belong to the set { + , ? , 0 } . An n -by- n sign 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 k 0 such that A k 0 for all k ≥ k 0 . 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 A n , 4 whose underlying graph G ( A n , 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 A n , 4 to be potentially eventually positive are established. All the minimal subpatterns of A n , 4 that allow an eventually positive matrix are identified. Consequently, all the potentially eventually positive subpatterns of A n , 4 are classified.
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