CompleteGraph

The complete graph $K_n$ — every pair of the $n$ vertices joined by an edge.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCompleteGraph✓↗

CompleteGraph(n)$K_n$, with vertices $1, \dots, n$.

Typed signatures

Type	Package	Applies
(integer) -> value	combinatorics	always

Domain: Collections

Examples

Basic #

$K_4$ has $\binom{4}{2} = 6$ edges

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In$\mathrm{EdgeCount}(\mathrm{CompleteGraph}(4))$Out$6$

Scope #

Every pair of the 4 vertices is joined

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In$\mathrm{CompleteGraph}(4)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(3, 4)\bigr\rbrack)$

$K_5$ has $\binom52=10$ edges

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In$\mathrm{CompleteGraph}(5)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(2, 5), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(3, 5), \mathrm{UndirectedEdge}(4, 5)\bigr\rbrack)$

Applications #

Every vertex reaches all others in one step

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In$\mathrm{Map}(()↦\mathrm{VertexEccentricity}(\mathrm{CompleteGraph}(5), \operatorname{\_1}), \mathrm{VertexList}(\mathrm{CompleteGraph}(5)))$Out$\bigl\lbrack1, 1, 1, 1, 1\bigr\rbrack$

Properties #

$K_6$ has 15 edges

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In$\mathrm{CompleteGraph}(6)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(1, 6), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(2, 5), \mathrm{UndirectedEdge}(2, 6), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(3, 5), \mathrm{UndirectedEdge}(3, 6), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(4, 6), \mathrm{UndirectedEdge}(5, 6)\bigr\rbrack)$

A triangle is both $K_3$ and the 3-cycle

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In$\bigl\lbrack\mathrm{CompleteGraph}(3), \mathrm{CycleGraph}(3)\bigr\rbrack$Out$\bigl\lbrack\mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(2, 3)\bigr\rbrack), \mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 1)\bigr\rbrack)\bigr\rbrack$

The wheel with 4 vertices is $K_4$

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In$\bigl\lbrack\mathrm{CompleteGraph}(4), \mathrm{WheelGraph}(4)\bigr\rbrack$Out$\bigl\lbrack\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(3, 4)\bigr\rbrack), \mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(4, 2)\bigr\rbrack)\bigr\rbrack$

The line graph of a 6-edge star is $K_6$

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In$\bigl\lbrack\mathrm{CompleteGraph}(6), \mathrm{LineGraph}(\mathrm{StarGraph}(7))\bigr\rbrack$Out$\bigl\lbrack\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(1, 6), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(2, 5), \mathrm{UndirectedEdge}(2, 6), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(3, 5), \mathrm{UndirectedEdge}(3, 6), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(4, 6), \mathrm{UndirectedEdge}(5, 6)\bigr\rbrack), \mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(1, 6), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(2, 5), \mathrm{UndirectedEdge}(2, 6), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(3, 5), \mathrm{UndirectedEdge}(3, 6), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(4, 6), \mathrm{UndirectedEdge}(5, 6)\bigr\rbrack)\bigr\rbrack$

About

Examples	8
Checked against	Wolfram Language 8 agree
Crosswalk	Wolfram Language

See also: PathGraph, CycleGraph, StarGraph