Hamming graph

Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H(d,q) has vertex set S^d, the set of ordered d-tuples of elements of S, or sequences of length d from S. Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph H(d,q) is, equivalently, the Cartesian product of d complete graphs K_q.

Hamming graph
Named after	Richard Hamming
Vertices	qd
Edges	${\displaystyle {\frac {d(q-1)q^{d}}{2}}}$
Diameter	d
Spectrum	${\displaystyle \{(d(q-1)-qi)^{{\binom {d}{i}}(q-1)^{i}};}$${\displaystyle i=0,\ldots ,d\}}$
Properties	d(q – 1)-regular Vertex-transitive Distance-regular Distance-balanced
Notation	H(d,q)
Table of graphs and parameters

In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes. Unlike the Hamming graphs H(d,q), the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.