There is a well-known relation between the spectrum of graph laplacian and its complement's laplacian, namely

$$λ_j (G^c) + λ_{n+2−j} (G) = n\;,$$

where the eigenvalues $λ_j$ are sorted in increasing order ($λ_1$ is always 0), $G^c$ is the graph's complement, and $n$ is the number of vertices. This can be easily proved by making use of the fact that the eigenvectors of symmetric matrices are orthogonal and the following relation

$$L(G) + L(G^c) = nI - J\;,$$

where $J$ is the matrix of ones.

This argument breaks down when we consider directed networks (digraphs), whose edges become directed and the laplacian matrix is no longer symmetric. Yet I found numerically that the relation $λ_j (G^c) + λ_{n+2−j} (G) = n$ still holds for digraphs, when we sort the eigenvalues according to their real part.

Is anyone aware of a proof or counterexample of the above claim?