Topological reduction of tight-binding models on branched networks

Noninteracting excitations on a generic discrete structure are described by means of a single-band tight-binding Hamiltonian. Despite the simplicity of the model, topological inhomogeneity gives rise to nontrivial spectral features. At the same time the lack of regularities makes the spectral analysis a rather hard task, since powerful tools such as Fourier transform are ruled out, In this paper we introduce a rigorous analytical technique, based on topological methods, for the exact solution of this problem in the case of branched structures. We provide some examples illustrating the technique.