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| | Now, one of [http://www.crunchbase.com/organization/chiropractor-orlando http://www.crunchbase.com/organization/chiropractor-orlando] the departed. For every type of person orlando fl chiropractor attending funeral. Howard says he was starting to learn how to plant a memorial service, pet memorials. All paperwork regarding the funeral parlor that may be too difficult as setting up the responsibility of taking the decision that is not being done. While their clients that they wish a tree to be gathered through the weekend. Each year, Mr. Pallbearers will be in the community wasn't as integrated elsewhere around White Plains, and all members of the funeral home.<br><br>My blog post; [http://www.crunchbase.com/organization/chiropractor-orlando http://www.crunchbase.com/organization/chiropractor-orlando] |
| | name = Complete graph
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| | image = [[Image:Complete graph K7.svg|200px]]
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| | image_caption = {{math|''K''<sub>7</sub>}}, a complete graph with 7 vertices
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| | vertices = {{mvar|n}}
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| | radius = <math>\left\{\begin{array}{ll}0 & n \le 1\\ 1 & \text{otherwise}\end{array}\right.</math>
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| | diameter = <math>\left\{\begin{array}{ll}0 & n \le 1\\ 1 & \text{otherwise}\end{array}\right.</math>
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| | girth = <math>\left\{\begin{array}{ll}\infty & n \le 2\\ 3 & \text{otherwise}\end{array}\right.</math>
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| | edges = <math>\textstyle\frac{n (n-1)}{2}</math>
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| |notation = {{math|''K<sub>n</sub>''}}
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| | automorphisms = {{math|''n''! ([[Symmetric group|''S'']]<sub>''n''</sub>)}}
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| | chromatic_number = {{mvar|n}}
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| | chromatic_index = {{mvar|n}} if {{mvar|n}} is odd<br>{{math|''n'' − 1}} if {{mvar|n}} is even
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| | spectrum = <math>\left\{\begin{array}{lll}\emptyset & n = 0\\\{0^1\} & n = 1\\ \{(n - 1)^1, -1^{n - 1}\} & \text{otherwise}\end{array}\right.</math><!-- is n = 1 really necessary? a partial case of the variant “otherwise”, isn’t it? -->
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| | properties = [[Regular graph|{{math|(''n'' − 1)}}-regular]]<br/>[[Symmetric graph]]<br/>[[Vertex-transitive graph|Vertex-transitive]]<br/>[[Edge-transitive graph|Edge-transitive]]<br/>[[strongly regular graph|Strongly regular]]<br/>[[Integral graph|Integral]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], a '''complete graph''' is a [[simple graph|simple]] [[undirected graph]] in which every pair of distinct [[vertex (graph theory)|vertices]] is connected by a unique [[edge (graph theory)|edge]]. A '''complete digraph''' is a [[directed graph]] in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).
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| Graph theory itself is typically dated as beginning with [[Leonhard Euler]]'s 1736 work on the [[Seven Bridges of Königsberg]]. However, [[graph drawing|drawing]]s of complete graphs, with
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| their vertices placed on the points of a [[regular polygon]], appeared already in the 13th century, in the work of [[Ramon Llull]].<ref>{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|authorlink=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.
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| </ref> Such a drawing is sometimes referred to as a '''mystic rose'''.<ref>{{citation|url=http://nrich.maths.org/6703|title=Mystic Rose | publisher=nrich.maths.org |accessdate=23 January 2012}}.</ref>
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| ==Properties==
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| The complete graph on {{mvar|n}} vertices is denoted by {{math|''K<sub>n</sub>''}}. Some sources claim that the letter K in this notation stands for the German word ''komplett'',<ref>{{citation|first1=David|last1=Gries|author1-link=David Gries|first2=Fred B.|last2=Schneider|author2-link=Fred B. Schneider|title=A Logical Approach to Discrete Math|publisher=Springer-Verlag|year=1993|page=436|url=http://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA436}}.</ref> but the German name for a complete graph, ''vollständiger Graph'', does not contain the letter K, and other sources state that the notation honors the contributions of [[Kazimierz Kuratowski]] to graph theory.<ref>{{citation|title=Mathematics All Around|first=Thomas L.|last=Pirnot|publisher=Addison Wesley|year=2000|isbn=9780201308150|page=154}}.</ref>
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| ''K''<sub>''n''</sub> has {{math|''n''(''n'' − 1)/2}} edges (a [[triangular number]]), and is a [[regular graph]] of [[degree (graph theory)|degree]] {{math|''n'' − 1}}. All complete graphs are their own [[Clique (graph theory)|maximal cliques]]. They are maximally [[connectivity (graph theory)|connected]] as the only [[vertex cut]] which disconnects the graph is the complete set of vertices. The [[complement graph]] of a complete graph is an [[empty graph]].
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| If the edges of a complete graph are each given an [[Orientation (graph theory)|orientation]], the resulting [[directed graph]] is called a [[tournament (graph theory)|tournament]].
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| The number of [[Matching (graph theory)|matchings]] of the complete graphs are given by the [[Telephone number (mathematics)|telephone numbers]]
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| :1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... {{OEIS|A000085}}.
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| These numbers give the largest possible value of the [[Hosoya index]] for an ''n''-vertex graph.<ref>{{citation
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| | last1 = Tichy | first1 = Robert F.
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| | last2 = Wagner | first2 = Stephan
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| | doi = 10.1089/cmb.2005.12.1004
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| | issue = 7
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| | journal = [[Journal of Computational Biology]]
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| | pages = 1004–1013
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| | title = Extremal problems for topological indices in combinatorial chemistry
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| | url = http://www.math.tugraz.at/fosp/pdfs/tugraz_main_0052.pdf
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| | volume = 12
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| | year = 2005}}.</ref> The number of [[perfect matching]]s of the complete graph ''K''<sub>''n''</sub> (with ''n'' even) is given by the [[double factorial]] (''n'' − 1)!!.<ref>{{citation|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|arxiv=0906.1317|year=2009}}.</ref>
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| The [[Crossing number (graph theory)|crossing numbers]] up to ''K''<sub>''27''</sub> are known, with ''K''<sub>''28''</sub> requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.<ref>{{cite web | url = //www.ist.tugraz.at/staff/aichholzer/research/rp/triangulations/crossing/ | title = Rectilinear Crossing Number project | author = Oswin Aichholzer}}</ref> Crossing numbers for ''K''<sub>''5''</sub> through ''K''<sub>''18''</sub> are
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| :1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, ... {{OEIS|A014540}}.
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| ==Geometry and topology==
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| A complete graph with {{mvar|n}} nodes represents the edges of an {{math|(''n'' − 1)}}-[[simplex]]. Geometrically {{math|''K''<sub>3</sub>}} forms the edge set of a [[triangle]], {{math|''K''<sub>4</sub>}} a [[tetrahedron]], etc. The [[Császár polyhedron]], a nonconvex polyhedron with the topology of a [[torus]], has the complete graph {{math|''K''<sub>7</sub>}} as its [[skeleton (topology)|skeleton]]. Every [[neighborly polytope]] in four or more dimensions also has a complete skeleton.
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| {{math|''K''<sub>1</sub>}} through {{math|''K''<sub>4</sub>}} are all [[planar graph]]s. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph {{math|''K''<sub>5</sub>}} plays a key role in the characterizations of planar graphs: by [[Kuratowski's theorem]], a graph is planar if and only if it contains neither {{math|''K''<sub>5</sub>}} nor the [[complete bipartite graph]] {{math|''K''<sub>3,3</sub>}} as a subdivision, and by [[Wagner's theorem]] the same result holds for [[graph minor]]s in place of subdivisions. As part of the [[Petersen family]], {{math|''K''<sub>6</sub>}} plays a similar role as one of the [[forbidden minor]]s for [[linkless embedding]].<ref>{{citation
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| | last1 = Robertson | first1 = Neil | author1-link = Neil Robertson (mathematician)
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| | last2 = Seymour | first2 = P. D. | author2-link = Paul Seymour (mathematician)
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| | last3 = Thomas | first3 = Robin | author3-link = Robin Thomas (mathematician)
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| | doi = 10.1090/S0273-0979-1993-00335-5
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| | arxiv = math/9301216 | mr = 1164063
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| | issue = 1
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| | journal = Bulletin of the American Mathematical Society
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| | pages = 84–89
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| | title = Linkless embeddings of graphs in 3-space
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| | volume = 28
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| | year = 1993}}.</ref>
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| In other words, and as Conway and Gordon<ref>{{cite journal |authorlink1=J. H. Conway|authorlink2=Cameron Gordon (mathematician)|author1=Conway, J. H. |author2=Cameron Gordon|title=Knots and Links in Spatial Graphs |journal=J. Graph Th. |volume=7 |issue=4 |pages=445–453 |year=1983 |doi=10.1002/jgt.3190070410}}</ref> proved, every embedding of {{math|''K''<sub>6</sub>}} is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any embedding of {{math|''K''<sub>7</sub>}} contains a knotted [[Hamiltonian cycle]].
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| ==Examples==
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| Complete graphs on {{mvar|n}} vertices, for {{mvar|n}} between 1 and 12, are shown below along with the numbers of edges:
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| {|class="wikitable"
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| ! {{math|''K''<sub>1</sub>: 0}} || {{math|''K''<sub>2</sub>: 1}} || {{math|''K''<sub>3</sub>: 3}} || {{math|''K''<sub>4</sub>: 6}}
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| |-
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| | [[Image:Complete graph K1.svg|140px]]
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| | [[Image:Complete graph K2.svg|140px]]
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| | [[Image:Complete graph K3.svg|140px]]
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| | [[Image:3-simplex graph.svg|140px]]
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| |-
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| ! {{math|''K''<sub>5</sub>: 10}} || {{math|''K''<sub>6</sub>: 15}} || {{math|''K''<sub>7</sub>: 21}} || {{math|''K''<sub>8</sub>: 28}}
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| |-
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| | [[Image:4-simplex graph.svg|140px]]
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| | [[Image:5-simplex graph.svg|140px]]
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| | [[Image:6-simplex graph.svg|140px]]
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| | [[Image:7-simplex graph.svg|140px]]
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| |-
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| ! {{math|''K''<sub>9</sub>: 36}} || {{math|''K''<sub>10</sub>: 45}} || {{math|''K''<sub>11</sub>: 55}} || {{math|''K''<sub>12</sub>: 66}}
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| |-
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| | [[Image:8-simplex graph.svg|140px]]
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| | [[Image:9-simplex graph.svg|140px]]
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| | [[Image:10-simplex graph.svg|140px]]
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| | [[Image:11-simplex graph.svg|140px]]
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| |-
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| |}
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| ==See also==
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| * [[Complete bipartite graph]]
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| * [[Shield of the Trinity]] (traditional Christian symbol which is a tetrahedral graph)
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{Wiktionary|complete graph}}
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| * {{MathWorld | urlname = CompleteGraph | title = Complete Graph}}
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| [[Category:Parametric families of graphs]]
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| [[Category:Regular graphs]]
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My blog post; http://www.crunchbase.com/organization/chiropractor-orlando