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 be a set of elements and a positive integer. The Hamming graph has vertex set , the set of ordered -tuples of elements of , or sequences of length from . Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph is, equivalently, the Cartesian product of complete graphs .

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 , the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.

Special cases

• , which is the generalized quadrangle

• , which is the complete graph

• , which is the lattice graph and also the rook's graph

• , which is the singleton graph

• , which is the hypercube graph . Hamiltonian paths in these graphs form Gray codes.

• Because Cartesian products of graphs preserve the property of being a unit distance graph, the Hamming graphs and are all unit distance graphs.

Applications

The Hamming graphs are interesting in connection with error-correcting codes and association schemes, to name two areas. They have also been considered as a communications network topology in distributed computing.

Computational complexity

It is possible in linear time to test whether a graph is a Hamming graph, and in the case that it is, find a labeling of it with tuples that realizes it as a Hamming graph.

References

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