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| | Hi there, I am Felicidad Oquendo. Interviewing is what she does but quickly she'll be on her personal. Arizona is her birth place and she will by no means move. The preferred hobby for my kids and me is playing crochet and now I'm trying to make cash with it.<br><br>Feel free to surf to my page - auto warranty, [http://Beyoufirst.org/ActivityFeed/MyProfile/tabid/62/userId/31300/Default.aspx Click To See More], |
| | name = Hamming graph
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| | namesake = [[Richard Hamming]]
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| | vertices = <math>q^d</math>
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| | edges = <math> \frac{d(q-1)q^d}{2}</math>
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| | diameter = <math>d</math>
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| | spectrum = <math>\{(d (q - 1) - q i)^{\binom{d}{i} (q - 1)^i}; i = 0, \ldots, d\}</math>
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| | properties = [[Regular graph|<math>d(q-1)</math>-regular]]<br/>[[Vertex-transitive graph|Vertex-transitive]]<br/>[[Distance-regular graph|Distance-regular]]<ref name="bh12"/>
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| | notation = H<math>(d,q)</math>
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| }}
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| '''Hamming graphs''' are a special class of [[graph (mathematics)|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 ''S<sup>d</sup>'', the set of ordered ''d''-tuples of elements of ''S'', or sequences of length ''d'' from ''S''. Two vertices are [[graph (mathematics)|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 graphs|Cartesian product]] of ''d'' [[complete graph]]s ''K''<sub>''q''</sub>.<ref name="bh12">{{citation
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| | last1 = Brouwer | first1 = Andries E. | author1-link = Andries Brouwer
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| | last2 = Haemers | first2 = Willem H.
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| | doi = 10.1007/978-1-4614-1939-6
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| | isbn = 978-1-4614-1938-9
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| | location = New York
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| | mr = 2882891
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| | page = 178
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| | publisher = Springer
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| | series = Universitext
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| | title = Spectra of graphs
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| | year = 2012}}.</ref>
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| In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.<ref name="ik00">{{citation
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| | last1 = Imrich | first1 = Wilfried
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| | last2 = Klavžar | first2 = Sandi | author2-link = Sandi Klavžar
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| | contribution = Hamming graphs
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| | isbn = 0-471-37039-8
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| | mr = 1788124
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| | pages = 104–106
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| | publisher = Wiley-Interscience, New York
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| | series = Wiley-Interscience Series in Discrete Mathematics and Optimization
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| | title = Product graphs
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| | year = 2000}}.</ref> Unlike the Hamming graphs ''H''(''d'',''q''), the graphs in this more general class are not necessarily [[distance-regular graph|distance-regular]], but they continue to be [[regular graph|regular]] and [[vertex-transitive graph|vertex-transitive]].
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| ==Special Cases==
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| *''H''(2,3), which is the generalized quadrangle ''G'' ''Q'' (2,1)<ref>{{citation
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| | last1 = Blokhuis | first1 = Aart
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| | last2 = Brouwer | first2 = Andries E. | author2-link = Andries Brouwer
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| | last3 = Haemers | first3 = Willem H.
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| | doi = 10.1007/s10623-007-9100-7
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| | issue = 1-3
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| | journal = Designs, Codes and Cryptography
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| | mr = 2336413
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| | pages = 293–305
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| | title = On 3-chromatic distance-regular graphs
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| | volume = 44
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| | year = 2007}}. See in particular note (e) on p. 300.</ref>
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| *''H''(1,''q''), which is the complete graph ''K''<sub>''q''</sub><ref name="dc04">{{citation
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| | last1 = Dekker | first1 = Anthony H.
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| | last2 = Colbert | first2 = Bernard D.
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| | contribution = Network robustness and graph topology
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| | location = Darlinghurst, Australia, Australia
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| | pages = 359–368
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| | publisher = Australian Computer Society, Inc.
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| | series = ACSC '04
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| | title = Proceedings of the 27th Australasian conference on Computer science - Volume 26
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| | url = http://dl.acm.org/citation.cfm?id=979922.979965
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| | year = 2004}}.</ref>
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| *''H''(2,''q''), which is the lattice graph ''L''<sub>''q,q''</sub> and also the [[rook's graph]]<ref>{{citation
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| | last1 = Bailey | first1 = Robert F.
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| | last2 = Cameron | first2 = Peter J.
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| | doi = 10.1112/blms/bdq096
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| | issue = 2
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| | journal = Bulletin of the London Mathematical Society
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| | mr = 2781204
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| | pages = 209–242
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| | title = Base size, metric dimension and other invariants of groups and graphs
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| | volume = 43
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| | year = 2011}}.</ref>
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| *''H''(''d'',1), which is the singleton graph ''K''<sub>1</sub>
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| *''H''(''d'',2), which is the [[hypercube graph]] ''Q''<sub>''d''</sub><ref name="bh12"/>
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| ==Applications==
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| The Hamming graphs are interesting in connection with [[error-correcting codes]]<ref>{{citation
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| | last = Sloane | first = N. J. A. | author-link = Neil Sloane
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| | journal = Graph Theory Notes of New York
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| | pages = 11–20
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| | title = Unsolved problems in graph theory arising from the study of codes
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| | url = http://neilsloane.com/doc/pace2.pdf
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| | volume = 18
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| | year = 1989}}.</ref> and [[association scheme]]s,<ref>{{citation
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| | last1 = Koolen | first1 = Jacobus H.
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| | last2 = Lee | first2 = Woo Sun
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| | last3 = Martin | first3 = William J.
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| | contribution = Characterizing completely regular codes from an algebraic viewpoint
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| | doi = 10.1090/conm/531/10470
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| | location = Providence, RI
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| | mr = 2757802
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| | pages = 223–242
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| | publisher = Amer. Math. Soc.
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| | series = Contemp. Math.
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| | title = Combinatorics and graphs
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| | volume = 531
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| | year = 2010}}. 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".</ref> to name two areas. They have also been considered as a communications network topology in [[distributed computing]].<ref name="dc04"/>
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| ==Computational complexity==
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| It is possible 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, in [[linear time]].<ref name="ik00"/>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{mathworld | title = Hamming Graph | id = HammingGraph}}
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| *{{cite web
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| | url = http://www.win.tue.nl/~aeb/graphs/Hamming.html
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| | title = Hamming graphs
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| | first = Andries E.
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| | last = Brouwer
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| | authorlink = Andries E. Brouwer
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| }}
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| [[Category:Parametric families of graphs]]
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| [[Category:Regular graphs]]
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Hi there, I am Felicidad Oquendo. Interviewing is what she does but quickly she'll be on her personal. Arizona is her birth place and she will by no means move. The preferred hobby for my kids and me is playing crochet and now I'm trying to make cash with it.
Feel free to surf to my page - auto warranty, Click To See More,