Spectra of the quotient of a directed graph Given a graph $G(V,E)$ and a partition $\{V_1,\dots V_n\}$ of the nodes set $V$, the adjacency and Laplacian spectra of the quotient graph $Q(G)$ interlaces the adjacent and the Laplacian spectra of the parent graph $G$. Besides, if the partition in equitable the interlacing is tight
This is a well known result.
Now, suppose I have a directed graph $D(V,E)$ and a partition $\{V_1,\dots V_n\}$ of the nodes set. Suppose also that the quotient graph $Q(D)$ is undirected. Under what conditions the interlacing and inclusion results for the adjacency and/or Laplacian spectra are still valid?
 A: Here are some comments.


*

*Perhaps you want to start with the question of conditions on a square $0,1$ matrix which assure that the spectrum is real, since you say that is a precondition for the  stronger thing you want to make sense.

*You probably want to assume the graph is strongly connected and aperiodic (  the directed cycles are not all of length $mk$ for some $m \gt 1.)$

*Certainly the largest eigenvalue is real and is especially interesting. To avoid the issue of complex eigenvalues, start with the question of when the quotient graph has smaller Peron eigenvalue.

*The example I gave (which fails the aperiodic requirement I suggest below) also fails this condition. If you consider a matrix where the $i,j$ entry is the probability of moving from vertex $i$ to vertex $j$ then the problem may go away.

*Undirected multigraphs (with loops) should preserve what you want with the right definition of quotient. Perhaps that is a useful pointer.

*I strongly suggest looking into the work of Fan Chung.
