  We propose a new localization result for the leading eigenvalue and
eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner
product between $A$ and a given rank-one landmark matrix $X$. Different choices
for $X$ may be used, depending upon the problem under investigation. In
particular, we show that the choice where $X$ is the all-ones matrix allows to
estimate the signature of the leading eigenvector of $A$, generalizing previous
results on Perron-Frobenius properties of matrices with some negative entries.
As another application we consider the problem of community detection in graphs
and networks. The problem is solved by means of modularity-based spectral
techniques, following the ideas pioneered by Miroslav Fiedler in mid 70s. We
show that a suitable choice of $X$ can be used to provide new quality
guarantees of those techniques, when the network follows a stochastic block
model.
