I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined contained at least one cycle, but this need not to be a necessary condition):

"If $\lambda$ is an eigenvalue of an adjacency matrix $A$, then it can be expressed as $r\cdot z$, where $r$ is some real number and z is $n$-th root of unity for some $n \in \mathbb{N}$. Moreover, if some eigenvalue $\lambda$ can be expressed in this form, then for all $n$-th roots of unity $z'$, $r \cdot z'$ is an eigenvalue of $A$. As a consequence, if $\lambda$ is eigenvalue of $A$, then also $|\lambda|$ (absolute value) is an eigenvalue of $A$."

Since I am not an expert in the field of spectral graph theory, I was unable to proof or disproof the property by myself. Is it known if this property or any similar property holds? Has anyone proved any similar property (it is possible, that the property does not hold exactly in the form I had written it down, but it may hold for instance for some special class of digraphs)? Any reference would be welcomed.

Thank you in advance.

rationaland $z$ is a primitive $n$th root of unity, then $rz'$ is also an eigenvalue whenever $z'$ is anotherprimitive$n$th root of unity. – Emil Jeřábek Sep 7 '11 at 13:22