Nearest-neighbor adjacency on Z^d - Imported Knowlpedia Content

On the lattice lattice-zd, two sites $x,y\in\mathbb{Z}^d$ are nearest neighbors (written $x\sim y$ ) if they differ by $1$ in exactly one coordinate and agree in all others.

Equivalently, using the $\ell^1$ norm,

x\sim y \quad\Longleftrightarrow\quad \|x-y\|_1 = 1,

where $\|z\|_1 := \sum_{i=1}^d |z_i|$ .

A convenient characterization is:

y = x \pm e_i \text{ for some } i\in\{1,\dots,d\},

where $e_i$ is the $i$ -th standard basis vector.

Remarks

Degree. Each site in $\mathbb{Z}^d$ has exactly $2d$ nearest neighbors.

This adjacency relation turns $\mathbb{Z}^d$ into an infinite graph, sometimes called the nearest-neighbor lattice graph.