I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph without diagonalizing it. The graphs I am interested in are not regular (but they have a maximum degree) or bipartite. They may or may not be planar.
- Are there any bounds for either of the extremal eigenvalues of the skew-adjacency matrix?
- Are there any bounds to the entries of the eigenvectors corresponding to the extremal eigenvalues that I can obtain without diagonalizing the skew-adjacency matrix?
- Suppose that I know that the extremal eigenvalues of the skew-adjacency matrix are degenerate. Does this tell me anything useful related to the above questions?