Graphical enumeration of polyhedral clusters

The following hypothesis is proposed: crystal structures may be ordered or classified according to the polymerization of those coordination polyhedra (not necessarily of the same type) with the higher bond valences. The linkage of polyhedra to form clusters is considered from a graph-theoretic viewpoint. Polyhedra are represented by the chromatic vertices of a (labelled) graph, in which different colours indicate coordination polyhedra of different type. The linking together of polyhedra is denoted by the presence of an edge or edges between vertices representing linked polyhedra, the number of edges between two vertices corresponding to the number of corners (atoms) common to both polyhedra. Information on geometrical isomerism is lost in this graphical representation, but the graphical characteristics are retained. The graph may be completely represented by its matrix, an n × n matrix [with (ⁿ₂) = N independent elements] denoting vertex linkage; it is convenient to represent the N independent matrix elements by the ordered set {a,b,c,...,N}. The collection of all permutations of the vertex labellings that preserve isomorphism is called the automorphism group Γ(G) of the graph. Γ(G) is a subgroup of the symmetric group S_n, and the complementary disjoint subgroup of S_n defines all distinct graphs whose vertex sets correspond to the (unordered) set {a,b,c,...,N}. However, it is more convenient in practice to work with the corresponding matrix-element symmetries that form a permutation group, denoted P. This particular formation allows the rigorous but natural distinction between graphical and geometrical isomers, and allows systematic investigation of their characteristics. Graphical isomers can be enumerated using Pólya's theorem, by substitution of permitted matrix elements as weight functions into the cycle index of the permutation group P, and can be derived as non-equivalent derangements of the integer set {a,b,c,...,N}. Geometrical isomers can be enumerated for a specific graphical isomer by successively applying Pólya's theorem to the distribution of shared elements over the total dement set of each polyhedron in turn, and can be derived in a similar fashion. The M₂(TO₄)₂φ_N clusters are considered as an example of this procedure.