Hamming graph

Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics 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 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. The Hamming graph H(d,q) is, equivalently, the Cartesian product of d complete graphs Kq.[1]

Hamming graph
Named afterRichard Hamming
Vertices
Edges
Diameter
Spectrum
Properties-regular
Vertex-transitive
Distance-regular[1]
Notation
Table of graphs and parameters
H(3,3) drawn as a unit distance graph

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

Special Cases
  • H(2,3), which is the generalized quadrangle G Q (2,1)[3]
  • H(1,q), which is the complete graph Kq[4]
  • H(2,q), which is the lattice graph Lq,q and also the rook's graph[5]
  • H(d,1), which is the singleton graph K1
  • H(d,2), which is the hypercube graph Qd.[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 H(d,2) and H(d,3) 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]

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References
  1. Brouwer, Andries E.; Haemers, Willem H. (2012), Spectra of graphs, Universitext, New York: Springer, p. 178, doi:10.1007/978-1-4614-1939-6, ISBN 978-1-4614-1938-9, MR 2882891.
  2. Imrich, Wilfried; Klavžar, Sandi (2000), "Hamming graphs", Product graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, pp. 104–106, ISBN 978-0-471-37039-0, MR 1788124.
  3. Blokhuis, Aart; Brouwer, Andries E.; Haemers, Willem H. (2007), "On 3-chromatic distance-regular graphs", Designs, Codes and Cryptography, 44 (1–3): 293–305, doi:10.1007/s10623-007-9100-7, MR 2336413. See in particular note (e) on p. 300.
  4. Dekker, Anthony H.; Colbert, Bernard D. (2004), "Network robustness and graph topology", Proceedings of the 27th Australasian conference on Computer science - Volume 26, ACSC '04, Darlinghurst, Australia, Australia: Australian Computer Society, Inc., pp. 359–368.
  5. Bailey, Robert F.; Cameron, Peter J. (2011), "Base size, metric dimension and other invariants of groups and graphs", Bulletin of the London Mathematical Society, 43 (2): 209–242, doi:10.1112/blms/bdq096, MR 2781204.
  6. Horvat, Boris; Pisanski, Tomaž (2010), "Products of unit distance graphs", Discrete Mathematics, 310 (12): 1783–1792, doi:10.1016/j.disc.2009.11.035, MR 2610282
  7. Sloane, N. J. A. (1989), "Unsolved problems in graph theory arising from the study of codes" (PDF), Graph Theory Notes of New York, 18: 11–20.
  8. Koolen, Jacobus H.; Lee, Woo Sun; Martin, William J. (2010), "Characterizing completely regular codes from an algebraic viewpoint", Combinatorics and graphs, Contemp. Math., 531, Providence, RI: Amer. Math. Soc., pp. 223–242, arXiv:0911.1828, doi:10.1090/conm/531/10470, ISBN 9780821848654, MR 2757802. 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".
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