This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc. 13 (2000),773--806. This is really a remarkable and beautiful theorem.
Douglas Lind
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