Godsil & Royle, Theorem 9.5.1: Extension for digraphs? I was wondering if there is an extension to digraphs for Theorem 9.5.1 in Godsil & Royle's Algebraic Graph Theory.
The Theorem can also be found in Willem Haemers paper Interlacing Eigenvalues and Graphs, http://arno.uvt.nl/show.cgi?fid=2980 , Theorem 2.1
Any leads would be appreciated. 
 A: The fundamental difficulty in extending interlacing to directed graphs is that "interlacing" has no meaning for complex numbers, we cannot say that an eigenvalue of a vertex-deleted subgraph $G\setminus u$ of $G$ lies 'between' two eigenvalues of $G$. So the short answer to the question is "no".
If $C$ is a directed cycle on $n$ vertices, then its characteristic polynomial is $t^n-1$ and its zeros are simple and distinct and are regularly distributed on the unit circle. If we delete vertex we get a directed path with characteristic polynomial $t^{n-1}$. This example shows that many consequences of interlacing in the real case cannot be extended in any obvious way.
For graphs, interlacing is equivalent to the fact that if $u\in V(G)$ then the poles of the rational function $\phi(G\setminus u,t)\phi(G,t)$ are simple, and the residues at the poles are positive. (Here $\phi$ denotes the characteristic polynomial.) For directed graphs, if $\phi(G\setminus u,t)\phi(G,t)$ has these properties then every zero of $\phi(G\setminus u,t)$ lies in the convex hull of
the zeros of $\phi(G,t)$, and this can be viewed as an analog of interlacing. (This a form of the Gauss-Lucas theorem.) Unfortunately these properties do not hold in general, although I do not have an example at hand.
In numerical analysis, there is something of a principle that if a matrix is not normal, the relation between its properties and its eigenvalue is much weaker than we expect, based on our experience with symmetric matrices. (Google on Trefethen and pseudospectra.)
