  The spectral properties of the adjacency matrix, in particular its largest
eigenvalue and the associated principal eigenvector, dominate many structural
and dynamical properties of complex networks. Here we focus on the localization
properties of the principal eigenvector in real networks. We show that in most
cases it is either localized on the star defined by the node with largest
degree (hub) and its nearest neighbors, or on the densely connected subgraph
defined by the maximum $K$-core in a $K$-core decomposition. The localization
of the principal eigenvector is often strongly correlated with the value of the
largest eigenvalue, which is given by the local eigenvalue of the corresponding
localization subgraph, but different scenarios sometimes occur. We additionally
show that simple targeted immunization strategies for epidemic spreading are
extremely sensitive to the actual localization set.
