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.



Complex Eigenvalues of Directed Graphs
Let $D$ be the Paley tournament on seven vertices. Its vertices are the integers mod seven
and there is an arc from $i$ to $j$ is $j-i$ is a non-zero square mod seven. The characteristic polynomial of the adjacency matrix is $(x-3)(x^2+x+2)^3$. The only real eigenvalue is 3, the remaining eigenvalues are equal to $(-1\pm\sqrt{-7})/2$, with absolute value $\sqrt{2}$. This can be generalized to Paley tournaments on $q$ vertices, where
$q\equiv3$ mod 4.


According to my computations (in sage) random directed graphs with seven vertices and arc probability 0.5 do well as counterexamples too.


