Hamming graph

Hamming graph
Hamming graph
Named after	Richard Hamming
Vertices	qd
Edges
Diameter	d
Spectrum
Properties	d(q – 1)-regular; Vertex-transitive; Distance-regular Distance-balanced
Notation	H(d,q)
	Table of graphs and parameters

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 $K q$ .^[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.

Special cases

$H (2,3)$ , which is the generalized quadrangle $G Q (2,1)$ ^[4]
$H (1, q)$ , which is the complete graph $K q$ ^[5]
$H (2, q)$ , which is the lattice graph $L q,q$ and also the rook's graph^[6]
$H (d,1)$ , which is the singleton graph $K 1$
$H (d,2)$ , which is the hypercube graph $Q d$ .^[1] Hamiltonian paths in these graphs form Gray codes.
Because Cartesian products of graphs preserve the property of being a unit distance graph,^[7] 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^[8] and association schemes,^[9] to name two areas. They have also been considered as a communications network topology in distributed computing.^[5]

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.^[3]

References

^ ^a ^b ^c Brouwer, Andries E.; Haemers, Willem H. (2012), "12.3.1 Hamming graphs" (PDF), 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, retrieved 2022-08-08.
^ Karami, Hamed (2022), "Edge distance-balanced of Hamming graphs", Journal of Discrete Mathematical Sciences and Cryptography, 25: 2667–2672, doi:10.1080/09720529.2021.1914363.
^ ^a ^b 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.
^ 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.
^ ^a ^b 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.
^ 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, S2CID 6684542.
^ 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
^ 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.
^ Koolen, Jacobus H.; Lee, Woo Sun; Martin, W (2010), "Characterizing completely regular codes from an algebraic viewpoint", Combinatorics and graphs, Contemp. Math., vol. 531, Providence, RI: Amer., pp. 223–242, arXiv:0911.1828, doi:10.1090/conm/531/10470, ISBN 9780821848654, MR 2757802, S2CID 8197351. 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".

External links

[bh12-1] Brouwer, Andries E.; Haemers, Willem H. (2012), "12.3.1 Hamming graphs" (PDF), 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, retrieved 2022-08-08.

[2] Karami, Hamed (2022), "Edge distance-balanced of Hamming graphs", Journal of Discrete Mathematical Sciences and Cryptography, 25: 2667–2672, doi:10.1080/09720529.2021.1914363.

[ik00-3] 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.

[4] 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.

[dc04-5] 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.

[6] 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, S2CID 6684542.

[7] 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

[8] 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.

[9] Koolen, Jacobus H.; Lee, Woo Sun; Martin, W (2010), "Characterizing completely regular codes from an algebraic viewpoint", Combinatorics and graphs, Contemp. Math., vol. 531, Providence, RI: Amer., pp. 223–242, arXiv:0911.1828, doi:10.1090/conm/531/10470, ISBN 9780821848654, MR 2757802, S2CID 8197351. 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".

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]