Several necessary or sufficient conditions for a sign pattern to allow eventual positivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T ) is related to the maximal multiplicity MaxMult(T ) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T ) of distinct eigenvalues over the symmetric matrices whose graphs are T ). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).

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