Symmetry properties of the Bethe and Husimi lattices

(Received 10 July 1992, revised 18 September 1992, accepted 6 October 1992)

Abstract
The traditional picture of the Bethe and Husimi lattices as structures embedded in a space of infinite dimensionality permits no account of their metric and symmetry properties. In a previous work, we followed the indication of Mosseri and Sadoc that such lattices can be embedded in two-dimensional hyperbolic spaces and studied their metric properties. In this work we use such a representation to identify the symmetries of the generic q-coordinated Bethe lattice and of its associated Husimi lattice and construct a generating algorithm that, besides the precise coordinates of the vertices, provides also a way of labelling them hierarchically. The addition of symmetry and of a metric background space to these lattices enrich their potentiality for use as substrate for model systems, since it permits a more detailed analytical treatment of the models studied.