  Complex molecules and mesoscopic structures are naturally described by
general networks of elementary building blocks and tight-binding is one of the
simplest quantum model suitable for studying the physical properties arising
from the network topology. Despite the simplicity of the model, topological
complexity can make the evaluation of the spectrum of the tight-binding
Hamiltonian a rather hard task, since the lack of translation invariance rules
out such a powerful tool as Fourier transform. In this paper we introduce a
rigorous analytical technique, based on topological methods, for the exact
solution of this problem on branched structures. Besides its analytic power,
this technique is also a promising engineering tool, helpful in the design of
netwoks displaying the desired spectral features.
