Periodic graph (geometry)

A periodic graph, in the context of geometry, is an infinite graph embedded in Euclidean space that exhibits translational symmetry. More specifically, a periodic graph is invariant under the action of a discrete translation group, meaning there exists a set of linearly independent translation vectors such that translating the entire graph by any linear combination of these vectors with integer coefficients leaves the graph unchanged.

The study of periodic graphs often appears in the fields of crystallography, material science, and computational chemistry, where the underlying structure of a crystal or material can be represented as a periodic graph with atoms corresponding to vertices and chemical bonds to edges.

Key characteristics of a periodic graph include:

• Infinite extent: Periodic graphs extend infinitely in at least one direction.
• Translational Symmetry: They possess a discrete translational symmetry group, which dictates how the graph repeats itself.
• Embedding: They are embedded in a Euclidean space (typically 2D or 3D).
• Unit Cell: A fundamental region, called the unit cell, exists such that translating it by vectors in the translation group generates the entire graph.
• Quotient Graph: The quotient of the periodic graph by its translation group is a finite graph representing the structure within the unit cell. This quotient graph provides a simplified representation of the infinite periodic structure.

The dimensionality of the translational symmetry group (i.e., the number of linearly independent translation vectors) determines the dimensionality of the periodic graph. For example, a 1D periodic graph (like a chain) is invariant under translation along a single direction, while a 3D periodic graph (like a crystal lattice) is invariant under translation along three independent directions.

The connectivity and structure of the periodic graph within the unit cell, combined with the translation group, fully defines the overall structure of the infinite graph. Properties of the material represented by the graph are often influenced by the specific arrangement of vertices and edges within the periodic structure.