Tree sign pattern that permit eventual positivity (Q2154296)

A square real matrix \(A\) is said to be eventually positive (EP) if there exists a least positive integer \(k_0\), called the index of \(A\), such that \(A^k > 0\) for all \(k \geq k_0\). It is known that a nonnegative \(n \times n\) matrix \(A\) is EP if and only if it is primitive and its index is no more than \((n - 1)^2 + 1\). However, an \(n \times n\) matrix may have some negative entries but being still EP, and in this case there is no bound on the index in terms of \(n\). A sign pattern is a square matrix with entries \(+, -, 0\). Denote by \(Q(\mathcal{A})\) the set of real matrices with sign pattern \(\mathcal{A}\). A sign pattern \(\mathcal{A}\) is called potentially eventually positive (PEP) if there is at least one EP matrix in \(Q(\mathcal{A})\). It is known that if the positive part of a sign pattern is primitive, then the sign pattern is PEP, but the converse of this statement is not true in general. The main result in this paper states that the converse holds for all tree sign patterns, where a tree sign pattern (TSP) is a sign pattern whose nonzero entries form a tree. In other words, if \(\mathcal{A}\) is a TSP, then \(\mathcal{A}\) is PEP if and only if its positive part is primitive. The authors further conjecture that the same result is true only for TSP's. They also determine the index of any \(2 \times 2\) matrix in terms of its entries and pose three questions at the end of the paper.