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.
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.)