  This paper deals with spectral graph theory issues related to questions of
monotonicity and comparison of eigenvalues. We consider finite directed graphs
with non symmetric edge weights and we introduce a special self-adjoint
operator as the sum of two non self-adjoint Laplacians. We investigate how the
perturbation of the graph can affect the eigenvalues. Our approach is to take
well known techniques from finite dimensional matrix analysis and show how they
can be generalized for graph Laplacians.
