The Hamming graph H(d,q) has vertex set Sd, 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.

[1] In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.

[3] 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.

The Hamming graphs are interesting in connection with error-correcting codes[8] and association schemes,[9] to name two areas.

They have also been considered as a communications network topology in distributed computing.