text: CompleteGraph - Maple Help
  
  GraphTheory
  
   CompleteGraph
   construct complete graph
  
   Calling Sequence CompleteGraph(n, opts) CompleteGraph(V, opts) CompleteGraph(n, m, opts) CompleteGraph(n1, n2,..., nk, opts)
  
  Parameters
  
   n, m - positive integers n1,...,nk - positive integers V - list of integers, strings or symbols (vertex labels) opts - (optional) one or more options as specified below
  
  Options
  
   The opts argument can contain one or more of the options shown below.
   • directed=true or false
   This option specifies whether the resulting graph should be directed. The default is false.
   • layout=true or false
   This option specifies whether a default coordinate assignment for the vertices of the graph should be made. The default is true.
   • selfloops=true or false
   This option specifies whether the generated graph should include a self-loop for each vertex. The default is false.
   • vertexpositions=p
   Specifies coordinate positions for the vertices for use with DrawGraph.
  
  Description
  
   • CompleteGraph(n) returns the complete graph on n vertices. CompleteGraph(V) does the same thing except the vertices are labeled using the entries of V.
   • CompleteGraph(n, m) returns the complete bipartite graph with bipartitions of size n and m.
   • CompleteGraph(n1,...,nk) returns the complete multipartite graph with partitions of size n1,..., nk.
  
  Examples
  
   > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
   > $\mathrm{K4}≔\mathrm{CompleteGraph}\left(4\right)$
   ${\mathrm{K4}}{≔}{\mathrm{Graph 1: an undirected graph with 4 vertices and 6 edge\left(s\right)}}$ (1)
   > $\mathrm{Edges}\left(\mathrm{K4}\right)$
   $\left\{\left\{{1}{,}{2}\right\}{,}\left\{{1}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{3}{,}{4}\right\}\right\}$ (2)
   > $\mathrm{DrawGraph}\left(\mathrm{K4}\right)$
   > $G≔\mathrm{CompleteGraph}\left(\left[a,b,c\right]\right)$
   ${G}{≔}{\mathrm{Graph 2: an undirected graph with 3 vertices and 3 edge\left(s\right)}}$ (3)
   > $\mathrm{Edges}\left(G\right)$
   $\left\{\left\{{a}{,}{b}\right\}{,}\left\{{a}{,}{c}\right\}{,}\left\{{b}{,}{c}\right\}\right\}$ (4)
   > $\mathrm{K23}≔\mathrm{CompleteGraph}\left(2,3\right)$
   ${\mathrm{K23}}{≔}{\mathrm{Graph 3: an undirected graph with 5 vertices and 6 edge\left(s\right)}}$ (5)
   > $\mathrm{Edges}\left(\mathrm{K23}\right)$
   $\left\{\left\{{1}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{1}{,}{5}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{2}{,}{5}\right\}\right\}$ (6)
   > $\mathrm{DrawGraph}\left(\mathrm{K23},\mathrm{style}=\mathrm{bipartite}\right)$
  
  Compatibility
  
   • The directed and layout options were introduced in Maple 2019.
   • The selfloops option was introduced in Maple 2020.
   • The GraphTheory[CompleteGraph] command was updated in Maple 2023.
   • The vertexpositions option was introduced in Maple 2023.
explanation: The provided text is a documentation extract for the `CompleteGraph` function in Maple, a mathematical software. It describes the function's purpose, parameters, options, and provides examples of its usage. The text does not contain a specific question or a question-and-answer pair.
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distilabel_metadata.content: CompleteGraph - Maple Help
  
  GraphTheory
  
   CompleteGraph
   construct complete graph
  
   Calling Sequence CompleteGraph(n, opts) CompleteGraph(V, opts) CompleteGraph(n, m, opts) CompleteGraph(n1, n2,..., nk, opts)
  
  Parameters
  
   n, m - positive integers n1,...,nk - positive integers V - list of integers, strings or symbols (vertex labels) opts - (optional) one or more options as specified below
  
  Options
  
   The opts argument can contain one or more of the options shown below.
   • directed=true or false
   This option specifies whether the resulting graph should be directed. The default is false.
   • layout=true or false
   This option specifies whether a default coordinate assignment for the vertices of the graph should be made. The default is true.
   • selfloops=true or false
   This option specifies whether the generated graph should include a self-loop for each vertex. The default is false.
   • vertexpositions=p
   Specifies coordinate positions for the vertices for use with DrawGraph.
  
  Description
  
   • CompleteGraph(n) returns the complete graph on n vertices. CompleteGraph(V) does the same thing except the vertices are labeled using the entries of V.
   • CompleteGraph(n, m) returns the complete bipartite graph with bipartitions of size n and m.
   • CompleteGraph(n1,...,nk) returns the complete multipartite graph with partitions of size n1,..., nk.
  
  Examples
  
   > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
   > $\mathrm{K4}≔\mathrm{CompleteGraph}\left(4\right)$
   ${\mathrm{K4}}{≔}{\mathrm{Graph 1: an undirected graph with 4 vertices and 6 edge\left(s\right)}}$ (1)
   > $\mathrm{Edges}\left(\mathrm{K4}\right)$
   $\left\{\left\{{1}{,}{2}\right\}{,}\left\{{1}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{3}{,}{4}\right\}\right\}$ (2)
   > $\mathrm{DrawGraph}\left(\mathrm{K4}\right)$
   > $G≔\mathrm{CompleteGraph}\left(\left[a,b,c\right]\right)$
   ${G}{≔}{\mathrm{Graph 2: an undirected graph with 3 vertices and 3 edge\left(s\right)}}$ (3)
   > $\mathrm{Edges}\left(G\right)$
   $\left\{\left\{{a}{,}{b}\right\}{,}\left\{{a}{,}{c}\right\}{,}\left\{{b}{,}{c}\right\}\right\}$ (4)
   > $\mathrm{K23}≔\mathrm{CompleteGraph}\left(2,3\right)$
   ${\mathrm{K23}}{≔}{\mathrm{Graph 3: an undirected graph with 5 vertices and 6 edge\left(s\right)}}$ (5)
   > $\mathrm{Edges}\left(\mathrm{K23}\right)$
   $\left\{\left\{{1}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{1}{,}{5}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{2}{,}{5}\right\}\right\}$ (6)
   > $\mathrm{DrawGraph}\left(\mathrm{K23},\mathrm{style}=\mathrm{bipartite}\right)$
  
  Compatibility
  
   • The directed and layout options were introduced in Maple 2019.
   • The selfloops option was introduced in Maple 2020.
   • The GraphTheory[CompleteGraph] command was updated in Maple 2023.
   • The vertexpositions option was introduced in Maple 2023.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a documentation extract for the `CompleteGraph` function in Maple, a mathematical software. It describes the function's purpose, parameters, options, and provides examples of its usage. The text does not contain a specific question or a question-and-answer pair.
  
  Label: no-question