Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is the combinatorial or bond topology of the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified as surgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.