Английская Википедия:Hamming graph
Шаблон:Short description Шаблон:Infobox 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 Шаблон:Mvar be a set of Шаблон:Mvar elements and Шаблон:Mvar a positive integer. The Hamming graph Шаблон:Math has vertex set Шаблон:Mvar, the set of ordered Шаблон:Mvar-tuples of elements of Шаблон:Mvar, or sequences of length Шаблон:Mvar from Шаблон:Mvar. Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph Шаблон:Math is, equivalently, the Cartesian product of Шаблон:Mvar complete graphs Шаблон:Mvar.[1]
In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.[2] Unlike the Hamming graphs Шаблон:Math, the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.
Special cases
- Шаблон:Math, which is the generalized quadrangle Шаблон:Math[3]
- Шаблон:Math, which is the complete graph Шаблон:Mvar[4]
- Шаблон:Math, which is the lattice graph Шаблон:Mvar and also the rook's graph[5]
- Шаблон:Math, which is the singleton graph Шаблон:Math
- Шаблон:Math, which is the hypercube graph Шаблон:Mvar.[1] Hamiltonian paths in these graphs form Gray codes.
- Because Cartesian products of graphs preserve the property of being a unit distance graph,[6] the Hamming graphs Шаблон:Math and Шаблон:Math are all unit distance graphs.
Applications
The Hamming graphs are interesting in connection with error-correcting codes[7] and association schemes,[8] to name two areas. They have also been considered as a communications network topology in distributed computing.[4]
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.[2]
References
External links
- ↑ 1,0 1,1 Шаблон:Citation.
- ↑ 2,0 2,1 Шаблон:Citation.
- ↑ Шаблон:Citation. See in particular note (e) on p. 300.
- ↑ 4,0 4,1 Шаблон:Citation.
- ↑ Шаблон:Citation.
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation.
- ↑ Шаблон:Citation. On p. 224, the authors write that "a careful study of completely regular codes in Hamming graphs is central to the study of association schemes".
