  We consider the normalized Laplace operator for directed graphs with positive
and negative edge weights. This generalization of the normalized Laplace
operator for undirected graphs is used to characterize directed acyclic graphs.
Moreover, we identify certain structural properties of the underlying graph
with extremal eigenvalues of the normalized Laplace operator. We prove
comparison theorems that establish a relationship between the eigenvalues of
directed graphs and certain undirected graphs. This relationship is used to
derive eigenvalue estimates for directed graphs. Finally we introduce the
concept of neighborhood graphs for directed graphs and use it to obtain further
eigenvalue estimates.
