Section
Discrete Structures
Graphs and lattice structures for statistical mechanics
A (simple, undirected) graph is an ordered pair where:
- is a set of vertices (or nodes).
- is a set of edges, where each edge is a 2-element subset .
If , then:
- and are the endpoints of the edge,
- and are adjacent (neighbors), and
- the edge is incident to each of its endpoints.
A graph-vertex-edge is finite if its vertex set is finite. For a simple graph, this implies is finite as well.
If and , then for a simple undirected graph,
Handshaking identity (useful fact). For a finite simple undirected graph,
Fix a positive integer . The integer lattice in dimension is
where denotes the integers.
Elements are often called lattice sites or lattice points.
A finite box (or finite cube) in the lattice lattice-zd is a finite region of the form
where is a nonnegative natural-numbers.
This is the cube centered at the origin with side length (in lattice units). Its cardinality is
Let be a finite subset of vertices in a graph. In the lattice setting, take with adjacency given by nearest-neighbor-zd.
Write if and are adjacent.
Outer (external) vertex boundary. The outer boundary of is
These are the vertices outside that are one step away from .
Inner (internal) vertex boundary. The inner boundary is
These are the vertices inside that have at least one neighbor outside.
Edge boundary. The edge boundary (also called the set of cut edges) is
On the lattice lattice-zd, two sites are nearest neighbors (written ) if they differ by in exactly one coordinate and agree in all others.
Equivalently, using the norm,
where .
A convenient characterization is:
where is the -th standard basis vector.