en>Khazar2: /* Properties of JB algebras */clean-up, typos fixed: it it → it is using AWB

2013-10-21T23:11:33Z

Properties of JB algebras: clean-up, typos fixed: it it → it is using AWB

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{{infobox graph
| name = Halved cube graph
| image = [[File:Demi-3-cube.svg|200px]]
| image_caption = The halved cube graph <math>\tfrac{1}{2}Q_3</math>
| vertices = 2''n-1''
| edges = ''n''(''n''-1)2''n''-3
| automorphisms = ''n''! 2''n-1''
| chromatic_number =
| chromatic_index =
| girth =
| diameter =
| spectrum =
| properties = [[Symmetric graph|Symmetric]] [[Distance regular graph|Distance regular]]
| notation = <math>\tfrac{1}{2}Q_n</math>
}}
[[File:CubeAndStel.svg|thumb|Construction of two demicubes (regular tetrahedra, forming a [[stella octangula]]) from a single cube. The halved cube graph of order three is the graph of vertices and edges of a single demicube. The halved cube graph of order four includes all of the cube vertices and edges, and all of the edges of the two demicubes.]]
In [[graph theory]], the '''halved cube graph''' or '''half cube graph''' of order ''n'' is the graph of the [[demihypercube]], formed by connecting pairs of vertices at distance exactly two from each other in the [[hypercube graph]]. This connectivity pattern produces two isomorphic graphs, disconnected from each other, each of which is the halved cube graph.

==Equivalent constructions==
The construction of the halved cube graph can be reformulated in terms of [[binary number]]s. The vertices of a hypercube may be labeled by binary numbers in such a way that two vertices are adjacent exactly when they differ in a single bit.
The demicube may be constructed from the hypercube as the [[convex hull]] of the subset of binary numbers with an even number of nonzero bits (the [[evil number]]s), and its edges connect pairs of numbers whose [[Hamming distance]] is exactly two.<ref>{{citation
| last1 = Indyk | first1 = Piotr | author1-link = Piotr Indyk
| last2 = Matoušek | first2 = Jiří | author2-link = Jiří Matoušek (mathematician)
| editor1-last = Goodman | editor1-first = Jacob E. | editor1-link = Jacob E. Goodman
| editor2-last = O'Rourke | editor2-first = Joseph | editor2-link = Joseph O'Rourke (professor)
| contribution = Low-distortion embeddings of finite metric spaces
| edition = 2nd
| isbn = 9781420035315
| page = 179
| publisher = CRC Press
| title = Handbook of Discrete and Computational Geometry
| url = http://books.google.com/books?hl=en&lr=&id=QS6vnl8WlnQC&oi=fnd&pg=PA179
| year = 2010}}.</ref>

It is also possible to construct the halved cube graph from a lower-order hypercube graph, without taking a subset of the vertices:
:<math>\frac{1}{2}Q_n = Q_{n-1}^2</math>
where the superscript 2 denotes the [[Graph power|square]] of the hypercube graph ''Q''''n'' − 1, the graph formed by connecting pairs of vertices whose distance is at most two in the original graph. For instance, the halved cube graph of order four may be formed from an ordinary three-dimensional cube by keeping the cube edges and adding edges connecting pairs of vertices that are on opposite corners of the same square face.

==Examples==
The halved cube graph <math>\tfrac{1}{2}Q_3</math> of order 3 is the [[complete graph]] ''K''4, the graph of the [[tetrahedron]]. The halved cube graph <math>\tfrac{1}{2}Q_4</math> of order 4 is ''K''2,2,2,2, the graph of the [[Convex regular 4-polytope|four-dimensional regular polytope]], the [[16-cell]]. The halved cube graph <math>\tfrac{1}{2}Q_5</math> of order five is sometimes known as the [[Clebsch graph]], and is the complement of the [[folded cube graph]] of order five which is more commonly called the Clebsch graph. It exists in the 5-dimensional [[uniform 5-polytope]], the [[5-demicube]].

==Properties==
Because it is the [[bipartite half]] of a [[distance-regular graph]], the halved cube graph is itself distance-regular.<ref>{{citation
| last1 = Chihara | first1 = Laura
| last2 = Stanton | first2 = Dennis
| doi = 10.1007/BF01788084
| issue = 2
| journal = Graphs and Combinatorics
| mr = 932118
| pages = 101–112
| title = Association schemes and quadratic transformations for orthogonal polynomials
| volume = 2
| year = 1986}}.</ref> And because it contains a hypercube as a [[spanning subgraph]], it inherits from the hypercube all monotone graph properties, such as the property of containing a [[Hamiltonian cycle]].

As with the hypercube graphs, and their [[Isometry|isometric]] (distance-preserving) subgraphs the [[partial cube]]s, a halved cube graph may be embedded isometrically into a [[real vector space]] with the [[Taxicab geometry|Manhattan metric]] (''L''1 distance function). The same is true for the isometric subgraphs of halved cube graphs, which may be recognized in [[polynomial time]]; this forms a key subroutine for an algorithm which tests whether a given graph may be embedded isometrically into a Manhattan metric.<ref>{{citation
| last1 = Deza | first1 = M. | author1-link = Michel Deza
| last2 = Shpectorov | first2 = S.
| doi = 10.1006/eujc.1996.0024
| issue = 2-3
| journal = European Journal of Combinatorics
| mr = 1379378
| pages = 279–289
| title = Recognition of the ''l''1-graphs with complexity ''O''(''nm''), or football in a hypercube
| volume = 17
| year = 1996}}.</ref>

For every halved cube graph of order five or more, it is possible to (improperly) color the vertices with two colors, in such a way that the resulting colored graph has no nontrivial symmetries. For the graphs of order three and four, four colors are needed to eliminate all symmetries.<ref>{{citation
| last1 = Bogstad | first1 = Bill
| last2 = Cowen | first2 = Lenore J.
| doi = 10.1016/j.disc.2003.11.018
| issue = 1-3
| journal = Discrete Mathematics
| mr = 2061481
| pages = 29–35
| title = The distinguishing number of the hypercube
| volume = 283
| year = 2004}}.</ref>

== Sequence ==
The two graphs shown are symmetric ''D''''n'' and ''B''''n'' [[Petrie polygon]] projections (2(''n'' − 1) and ''n'' [[dihedral symmetry]]) of the related polytope which can include overlapping edges and vertices.
{| class="wikitable"
!''n''
![[Polytope]]
!Graph
!Vertices
!Edges
|- align=center
!2
||[[Line segment]]||[[Image:Complete graph K2.svg|60px]]||2|| –
|- align=center
!3
|[[tetrahedron]]||[[File:3-demicube.svg|60px]][[File:3-demicube_t0_B3.svg|60px]]||4||6
|- align=center
!4
|[[16-cell]]||[[File:4-demicube_t0_D4.svg|60px]][[File:4-demicube_t0_B4.svg|60px]]||8||24
|- align=center
!5
|[[5-demicube]]||[[File:5-demicube_t0_D5.svg|60px]][[File:5-demicube_t0_B5.svg|60px]]||16||80
|- align=center
!6
|[[6-demicube]]||[[File:6-demicube_t0_D6.svg|60px]][[File:6-demicube_t0_B6.svg|60px]]||32||240
|- align=center
!7
|[[7-demicube]]||[[File:7-demicube_t0_D7.svg|60px]][[File:7-demicube_t0_B7.svg|60px]]||64||672
|- align=center
!8
|[[8-demicube]]||[[File:8-demicube_t0_D8.svg|60px]][[File:8-demicube_t0_B8.svg|60px]]||128||1792
|- align=center
!9
|[[9-demicube]]||[[File:9-demicube_t0_D9.svg|60px]][[File:9-demicube_t0_B9.svg|60px]]||256||4608
|- align=center
|10
|[[10-demicube]]|||[[File:10-demicube.svg|60px]][[File:10-demicube graph.png|60px]]||512||11520
|}

==References==
{{reflist}}

==External links==
*{{mathworld|id=HalvedCubeGraph|title=Halved Cube Graph}}

[[Category:Parametric families of graphs]]
[[Category:Regular graphs]]

Jordan operator algebra - Revision history

en>Khazar2: /* Properties of JB algebras */clean-up, typos fixed: it it → it is using AWB