I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph. The graphs I am interested in are not regular (but they have a maximum degree) or bipartite. They may or may not be planar.

1. Are there any bounds for either of the extremal eigenvalues of the skew-adjacency matrix?
2. Is there a way to obtain the eigenvector corresponding to either of the extremal eigenvalues without diagonalizing the skew-adjacency matrix?
3. Are there any known results that may help with either of the above?
4. Suppose that I know somehow that the largest eigenvalue of the skew-adjacency matrix is degenerate. Does this tell me anything useful related to the above questions?