This is true for quadratic polynomials with constant term +1. Any such polynomial is the determinant of a matrix in SL₂ ℤ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of [[1,1],[0,1]] and [[1,0],[1,1]] (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of this paper.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix - there might be other factors.

Added comment: The general quadratic case might be possible to work out using Markov partitions of the induced map of a torus.

I forgot about the cyclotomic case, which can occur if the matrix is not Perron-Frobenius. If the polynomial is irreducible, I think the condition implies that the maximal norm roots are complex Perron numbers (or cyclotomic). These crop up in work of Kenyon on self-similar tilings (MR1392326 (97j:52025) ).

This is true for quadratic polynomials with constant term $+1$. Any such polynomial is the determinant of a matrix in $SL_2(\mathbb{Z})$ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of $\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$ and $\left(\begin{matrix}1&0\\1&1\end{matrix}\right)$ (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of On canonical triangulations of once-punctured torus bundles and two-bridge link complements.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix—there might be other factors.

Added comment: The general quadratic case might be possible to work out using Markov partitions of the induced map of a torus.

I forgot about the cyclotomic case, which can occur if the matrix is not Perron-Frobenius. If the polynomial is irreducible, I think the condition implies that the maximal norm roots are complex Perron numbers (or cyclotomic). These crop up in work of Kenyon on self-similar tilings (The construction of self-similar tilings, MR1392326).

This is true for quadratic polynomials with constant term +1. Any such polynomial is the determinant of a matrix in SL₂ ℤ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of [[1,1],[0,1]] and [[1,0],[1,1]] (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of this paper.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix - there might be other factors.

This is true for quadratic polynomials with constant term +1. Any such polynomial is the determinant of a matrix in SL₂ ℤ (e.g. using the companion matrix as indicated by Ben Webster). It's well known that any such matrix is conjugate to a multiple product of [[1,1],[0,1]] and [[1,0],[1,1]] (upper and lower triangular unipotent matrices). I'm not sure the original reference for this fact, but a reference is Proposition 2.1 of this paper.

I believe that your criterion implies that the maximal root of the polynomial is a Perron number. If so, then Lind has shown that every Perron number occurs as the spectral radius of a non-negative integral Perron-Frobenius matrix (and therefore the spectral radius of a recurrent digraph). This only implies that the polynomial divides the characteristic polynomial of the matrix - there might be other factors.

Added comment: The general quadratic case might be possible to work out using Markov partitions of the induced map of a torus.

I forgot about the cyclotomic case, which can occur if the matrix is not Perron-Frobenius. If the polynomial is irreducible, I think the condition implies that the maximal norm roots are complex Perron numbers (or cyclotomic). These crop up in work of Kenyon on self-similar tilings (MR1392326 (97j:52025) ).