Nearest-neighbor adjacency on Z^d

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,

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

A convenient characterization is:

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

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.