# Analogue of Perron numbers for non-negative matrices

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.

You are looking for "weak Perron numbers", which are numbers such that some positive integer power is a Perron number. In your example $$\lambda=\sqrt{2}$$, its square $$\lambda^2=2$$ is a Perron number, being an eigenvalue of a square matrix whose entries are all $$2$$.
Theorem: $$x \in \mathbb R$$ is a weak Perron number if and only if $$x$$ is the dominant eigenvalue of some square matrix of non-negative integers.