Tight-binding models on branched structures

In this paper we analyze the properties of electrons in noncrystalline stru ctures, mathematically described by graphs. We consider a tight-binding mod el for noninteracting quantum particles and its perturbative expansion in t he hopping parameter, which can be mapped into a random-walk problem on the same graph. The model is solved on a wide class of structures, called bund led graphs, which are used as models for the geometrical structure of polym ers and are obtained joining to each point of a ''base'' graph a copy of a ''fiber'' graph. The analytical calculation of the Green's functions is obt ained through an exact resummation of the perturbative series using graph c ombinatorial techniques. In particular, our result shows that when the base graph is a d-dimensional crystalline lattice, the fibers generate a self-e nergy of pure geometrical origin in the base Green's functions.