If $A$ is a square matrix with positive integer coefficients with a biggest real eigenvalues $\lambda>1$, then $\lambda$ is a *Perron-number*, which means that it is a real algebraic integer bigger than $1$, whose Galois conjugates $\mu$ satisfy $\lvert \mu\rvert<\lambda$. This follows from the Perron-Frobenius theorem.

The result is no longer true if $A$ is a square matrix with non-negative integer coefficients, as the simple example $\begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}$ shows: here $\lambda=\sqrt{2}$, which is not a Perron-number.

Here we know that the Galois-conjugates $\mu$ satisfy $\lvert \mu\rvert \le\lambda$ and that those where equality holds are simply $\lambda\theta$ where $\theta$ is a root of unity.

Is there then an analogue of the Perron-numbers for the real algebraic integers $\lambda>1$ that are biggest real eigenvalues of a square matrix with non-negative integer coefficients? I would be happy to have another name of such algebraic integers, and possibly literature about these numbers.