nt.number theory - Perron-Frobenius "inverse eigenvalue problem"

The answer to a sharper question involving integers, rather than rationals, is affirmative.

Let $lambda$ be a positive real algebraic integer that is greater in absolute value than all its Galois conjugates ("Perron number" or "PF number"). Then $lambda$ is the Perron–Frobenius eigenvalue of a positive integer matrix.

(The converse statement is an integer version of the Perron–Frobenius theorem, and is easy to prove.)

In a slightly weaker form (aperiodic non-negative matrix), this is theorem of Douglas Lind, from

The entropies of topological Markov shifts and a related class of algebraic integers.
Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283--300 (MR)

I don't have a good reference for the strong form, but it was discussed at Thurston seminar in 2008-2009. One interesting thing to note is that, while the proof can be made constructive, it is non-uniform: the size of the matrix can be arbitrarily large compared to the degree of $lambda$.