335:
229:
105:
269:
480:
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
1069:. 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".
697:
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.
1122:
130:
819:
746:
1020:
Koolen, Jacobus H.; Lee, Woo Sun; Martin, W (2010), "Characterizing completely regular codes from an algebraic viewpoint",
326:
905:
Bailey, Robert F.; Cameron, Peter J. (2011), "Base size, metric dimension and other invariants of groups and graphs",
814:, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, pp. 104–106,
1047:
954:
56:
438:
355:
635:
461:
233:
893:, ACSC '04, Darlinghurst, Australia, Australia: Australian Computer Society, Inc., pp. 359–368
1127:
504:
293:
111:
674:
496:
297:
682:
410:
123:
37:
681:, to name two areas. They have also been considered as a communications network topology in
1057:
977:
926:
870:
829:
756:
8:
887:
Dekker, Anthony H.; Colbert, Bernard D. (2004), "Network robustness and graph topology",
639:
523:
345:
49:
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815:
742:
379:
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914:
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371:
1087:
888:
807:
789:
1105:
1053:
973:
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866:
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825:
752:
719:
614:
577:
359:
30:
723:
1039:
968:
890:
Proceedings of the 27th
Australasian conference on Computer science - Volume 26
468:
861:
738:
1116:
567:
500:
289:
334:
367:
1024:, Contemp. Math., vol. 531, Providence, RI: Amer., pp. 223–242,
918:
990:
694:
363:
847:; Haemers, Willem H. (2007), "On 3-chromatic distance-regular graphs",
994:
1092:
628:
995:"Unsolved problems in graph theory arising from the study of codes"
1030:
776:
Karami, Hamed (2022), "Edge distance-balanced of
Hamming graphs",
391:
441:
if they differ in precisely one coordinate; that is, if their
422:
495:, the graphs in this more general class are not necessarily
778:
Journal of
Discrete Mathematical Sciences and Cryptography
224:{\displaystyle \{(d(q-1)-qi)^{{\binom {d}{i}}(q-1)^{i}};}
1082:
842:
673:
The
Hamming graphs are interesting in connection with
236:
133:
59:
263:
223:
99:
733:, Universitext, New York: Springer, p. 178,
1114:
1019:
947:
718:
904:
886:
805:
188:
175:
952:(2010), "Products of unit distance graphs",
874:. See in particular note (e) on p. 300.
258:
134:
907:Bulletin of the London Mathematical Society
688:
1029:
967:
860:
333:
100:{\displaystyle {\frac {d(q-1)q^{d}}{2}}}
1115:
989:
882:
880:
801:
799:
775:
714:
712:
710:
1083:
16:Cartesian product of complete graphs
1101:
877:
796:
707:
13:
179:
14:
1139:
1076:
638:preserve the property of being a
510:
362:and used in several branches of
849:Designs, Codes and Cryptography
668:
592:, which is the singleton graph
264:{\displaystyle i=0,\ldots ,d\}}
1013:
1002:Graph Theory Notes of New York
983:
941:
898:
836:
769:
547:, which is the complete graph
327:Table of graphs and parameters
207:
194:
168:
155:
143:
137:
78:
66:
1:
1123:Parametric families of graphs
790:10.1080/09720529.2021.1914363
722:; Haemers, Willem H. (2012),
700:
664:are all unit distance graphs.
636:Cartesian products of graphs
7:
522:, which is the generalized
10:
1144:
969:10.1016/j.disc.2009.11.035
810:(2000), "Hamming graphs",
499:, but they continue to be
445:is one. The Hamming graph
862:10.1007/s10623-007-9100-7
739:10.1007/978-1-4614-1939-6
429:, or sequences of length
325:
304:
274:
122:
110:
48:
36:
26:
21:
1022:Combinatorics and graphs
689:Computational complexity
724:"12.3.1 Hamming graphs"
354:are a special class of
1040:10.1090/conm/531/10470
675:error-correcting codes
460:is, equivalently, the
348:
265:
225:
101:
683:distributed computing
642:, the Hamming graphs
627:in these graphs form
417:, the set of ordered
394:. The Hamming graph
337:
266:
226:
102:
955:Discrete Mathematics
437:. Two vertices are
234:
131:
57:
1103:Brouwer, Andries E.
919:10.1112/blms/bdq096
845:Brouwer, Andries E.
720:Brouwer, Andries E.
679:association schemes
640:unit distance graph
346:unit distance graph
1085:Weisstein, Eric W.
806:Imrich, Wilfried;
693:It is possible in
349:
261:
221:
97:
962:(12): 1783–1792,
821:978-0-471-37039-0
748:978-1-4614-1938-9
731:Spectra of graphs
625:Hamiltonian paths
505:vertex-transitive
462:Cartesian product
332:
331:
300:Distance-balanced
294:Vertex-transitive
186:
95:
1135:
1109:
1106:"Hamming graphs"
1098:
1097:
1070:
1068:
1033:
1017:
1011:
1009:
999:
991:Sloane, N. J. A.
987:
981:
980:
971:
945:
939:
937:
902:
896:
894:
884:
875:
873:
864:
855:(1–3): 293–305,
843:Blokhuis, Aart;
840:
834:
832:
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763:
728:
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663:
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497:distance-regular
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476:
467:
459:
443:Hamming distance
436:
432:
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389:
385:
377:
372:computer science
343:
321:
298:Distance-regular
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61:
44:
19:
18:
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1113:
1112:
1088:"Hamming Graph"
1079:
1074:
1073:
1050:
1018:
1014:
997:
988:
984:
950:Pisanski, Tomaž
948:Horvat, Boris;
946:
942:
903:
899:
885:
878:
841:
837:
822:
804:
797:
774:
770:
761:
759:
749:
726:
717:
708:
703:
691:
671:
654:
643:
621:
617:
615:hypercube graph
613:, which is the
603:
599:
593:
582:
574:
570:
566:, which is the
556:
552:
548:
537:
526:
516:
513:
481:
475:
471:
469:complete graphs
465:
446:
434:
430:
426:
425:of elements of
418:
414:
395:
387:
383:
375:
360:Richard Hamming
338:
308:
296:
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210:
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62:
60:
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31:Richard Hamming
17:
12:
11:
5:
1141:
1131:
1130:
1128:Regular graphs
1125:
1111:
1110:
1099:
1078:
1077:External links
1075:
1072:
1071:
1048:
1012:
982:
940:
913:(2): 209–242,
897:
876:
835:
820:
812:Product graphs
808:Klavžar, Sandi
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352:Hamming graphs
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787:
784:: 2667–2672,
783:
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732:
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721:
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713:
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589:
585:
581:
579:
576:and also the
569:
568:lattice graph
563:
559:
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519:
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511:Special cases
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760:, retrieved
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669:Applications
659:
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583:
578:rook's graph
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358:named after
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1117:Categories
762:2022-08-08
701:References
629:Gray codes
524:quadrangle
275:Properties
1093:MathWorld
1031:0911.1828
250:…
201:−
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634:Because
439:adjacent
305:Notation
124:Spectrum
112:Diameter
38:Vertices
1066:8197351
1058:2757802
1008:: 11–20
978:2610282
935:6684542
927:2781204
871:2336413
830:1788124
757:2882891
501:regular
392:integer
374:. Let
290:regular
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356:graphs
1062:S2CID
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