Elements of tight-binding method in terms of graph theory

(Received 29 April 1993, revised 19 July 1993, accepted 28 July 1993)

Abstract
A description of the diamond structure in terms of graph theory obtained by using an additional axis (similar to the additional axis one uses in the case of the hexagonal lattice) is presented. The group of all graph-automorphisms of the graph we use as model is isomorphic to the space group O (the symmetry group of the diamond structure) and it admits natural representations in some mathematical spaces that can be associated to the graph. These spaces allow us to obtain advantageous mathematical formalisms for certain models and offer new possibilities to extend them to crystals with impurities or defects, quasi-crystals, etc. We exemplify by considering some elements of the tight-binding method.