Maximal independence and symmetry in crystal chemistry of natural tectosilicates.

Löwenstein's avoidance rule in aluminosilicates is reinterpreted on the basis of the fourth Pauling rule. It is shown that avoidance of Si-O-Si bridges may account for avoidance of Al-O-Al bridges. In view of this interpretation, it is proposed that the most favourable distributions of cations entering in substitution of silicon in the framework are associated to maximal independent sets of the respective 3-periodic nets.

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.

Combinatorial aspects of the Löwenstein avoidance rule. Part II: local and global constraints.

The determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio; the respective motif defines a periodic factor of the net. This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus.

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial.

According to Löwenstein's rule, Al-O-Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net.

Vertex collisions in 3-periodic nets of genus 4.

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known.

Non-crystallographic nets: characterization and first steps towards a classification.

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T).

Non-crystallographic nets with freely acting, non-abelian local automorphism groups.

Non-crystallographic (NC) nets are defined as periodic nets whose automorphism groups are not isomorphic to any isometry group in Euclidean space. This work focuses on a simple class of NC nets, restricted to nets with non-abelian, freely acting local automorphism groups. A general method is presented to derive such NC nets from crystallographic nets and some non-trivial examples are explored.