Combinatorial aspects of the Löwenstein avoidance rule. Part III: the relational system of configurations

This paper introduces a new method of determining the independence ratio of periodic nets, based on the observation that, in any maximum independent set of the whole net, be it periodic or not, the vertices of every unit cell should constitute an independent set, called here a configuration. For 1-periodic graphs, a configuration digraph represents possible sequences of configurations of the unit cell along the periodic line. It is shown that maximum independent sets of the periodic graph are based on directed cycles with the largest ratio. In the case of 2-periodic nets, it is necessary to draw a different configuration digraph for each crystallographic direction defining a linkage between neighbouring cells, a concept known as a binary relational system. The two possible systems are analysed in this paper: \overrightarrow{\bf{sql}} is associated to nets displaying linkages between unit cells along the directions 10 and 01, and \overrightarrow{\bf{hxl}} is associated to nets also displaying linkages between cells along the direction 11. For both kinds of nets, a maximum independent set is obtained as a homomorphic image from \overrightarrow{\bf{sql}} or \overrightarrow{\bf{hxl}} to the respective configuration system. The method is illustrated with some of the 2-periodic nets listed on the Reticular Chemistry Structure Resource site; it is shown that it provides a rigorous solution to the case of the net sdh that was not satisfactorily solved in Part II [Moreira de Oliveira, de Abreu Mendes & Eon (2022). Acta Cryst. A78, 115-127]. The method is extended to relational systems based on non-translational symmetry operations. The successive steps are then summarized and a simple application to the 3-periodic net qtz is discussed; analysis of zeolites and aluminosilicates may proceed along the same lines. It is shown that the new method enables the analysis of disordered distributions in periodic nets.