Periodic nets and quotient graphs#

Crystal structures are often idealised as nets: periodic graphs in which vertices represent sites and edges represent connections between them. Here we formalise that idea and introduce the vector representation and quotient graph.

Periodic nets#

A net in the sense used here is:

  • a connected, simple graph (no loops, no multiple edges),

  • realised as a periodic structure in Euclidean space.

We say that a net is n-periodic if it has translational symmetry in exactly n linearly independent directions (for crystal structures of interest, n = 3). The repeat unit is typically taken to be a primitive unit cell.

It is important to distinguish:

  • n-periodic: the net has n independent translation vectors (e.g. a 3-periodic net fills 3D space with translational symmetry),

  • n-dimensional: the net admits a faithful embedding in n dimensions but not in n-1 dimensions.

A graph can be 2-dimensional (planar) and still be 3-periodic if it is embedded periodically in three-dimensional space, e.g. a layered material.

Vector representation of a periodic net#

For a periodic net, it is convenient to choose:

  • a primitive unit cell with basis vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\), and

  • a numbering of the vertices in one reference cell: \(1, 2, \dots, v\).

Every edge in the infinite periodic net can be represented as a tuple

\[(i, j, u, v, w),\]

meaning:

  • the edge connects vertex i in the reference cell to

  • vertex j in the cell shifted by the lattice vector \(u \mathbf{a} + v \mathbf{b} + w \mathbf{c}\).

The list of all such edge tuples within the primitive cell is called the vector representation of the net. For simple graphs we only store one tuple for each undirected edge.

This representation plays, for periodic nets, the same role that an adjacency matrix plays for finite graphs. In principle, all topological properties (ring sizes, connectivity, combinatorial symmetry, etc.) can be recovered from it.

Quotient graph#

The quotient graph compresses the infinite periodic net down to a finite labelled graph associated with a chosen unit cell:

  • The vertices of the quotient graph correspond to vertices in the reference cell.

  • Each edge of the quotient graph corresponds to an edge tuple \((i, j, u, v, w)\) in the vector representation.

  • The edge is labelled by the translation \((u, v, w)\).

For example, in the diamond net (dia) there are two vertices in the primitive cell (labelled 1 and 2) and four edges in the vector representation:

  • (1, 2, 0, 0, 0)

  • (1, 2, 1, 0, 0)

  • (1, 2, 0, 1, 0)

  • (1, 2, 0, 0, 1)

The quotient graph therefore has two vertices, with four edges between them carrying these labels. It is allowed to have loops and multiple edges.

Genus and minimal nets#

The cyclomatic number of the quotient graph is sometimes called the genus of the net. For a connected quotient graph with e edges and v vertices this is

\[g = 1 + e - v.\]

The minimum possible genus of an n-periodic net is n. Nets that achieve this minimum are called minimal nets. For example, a 3-periodic net must have genus at least 3; those with genus 3 are minimal in this sense.

Minimal nets and quotient graphs are useful both for classification and for constructing canonical representations of periodic nets, which in turn allows algorithms to test when two periodic nets are topologically equivalent.