This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# CompleteGraph

 CompleteGraph[n] gives the complete graph with n vertices . CompleteGraphgives the complete k-partite graph with vertices .
• CompleteGraph[n] gives a graph with n vertices and an edge between every pair of vertices.
• CompleteGraph gives a graph with vertices partitioned into disjoint sets with vertices each and edges between all vertices in different sets and , but no edges between vertices in the same set .
The first few complete graphs :
Bipartite graphs :
Use a list of length to generate a -partite graph:
Directed complete graphs use two directional edges for each undirected edge:
Directed complete -partite graphs use directed edges from one group to another:
The first few complete graphs :
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Bipartite graphs :
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Use a list of length to generate a -partite graph:
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Directed complete graphs use two directional edges for each undirected edge:
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Directed complete -partite graphs use directed edges from one group to another:
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 Scope   (1)
Evaluate for a large argument:
 Options   (77)
By default an undirected graph is generated:
Use DirectedEdges->True to generate a directed graph:
Generate directed -partite graphs:
Label the edge :
Label all edges individually:
Use any expression as a label:
Use Placed with symbolic locations to control label placement along an edge:
Use explicit coordinates to place labels:
Vary positions within the label:
Place multiple labels:
Use automatic labeling by values through Tooltip and StatusArea:
Get a list of built-in settings for EdgeShapeFunction:
Undirected edges including the basic line:
Lines with different glyphs on the edges:
Directed edges including solid arrows:
Line arrows:
Open arrows:
Specify an edge function for an individual edge:
Combine with a different default edge function:
Draw edges by running a program:
EdgeShapeFunction can be combined with EdgeStyle:
EdgeShapeFunction has higher priority than EdgeStyle:
Style all edges:
Style individual edges:
Specify a weight for all edges:
Use any numeric expression as a weight:
By default the layout is chosen automatically:
Specify layouts on special curves:
Specify layouts that satisfy optimality criteria:
VertexCoordinates overrides GraphLayout coordinates:
Use AbsoluteOptions to extract VertexCoordinates computed using a layout algorithm:
Highlight the vertex :
Highlight the edge :
Highlight the vertices and edges:
Get a list of built-in settings for GraphHighlightStyle:
Use built-in settings for GraphHighlightStyle:
Get a list of built-in settings for GraphStyle:
Use built-in settings for GraphStyle:
Specify a property for vertices:
Edges:
By default any vertex coordinates are computed automatically:
Extract the resulting vertex coordinates using AbsoluteOptions:
Specify a layout function along an ellipse:
Use it to generate vertex coordinates for a graph:
VertexCoordinates has higher priority than GraphLayout:
Use vertex names as labels:
Label individual vertices:
Label all vertices:
Use any expression as a label:
Use Placed with symbolic locations to control label placement, including outside positions:
Symbolic outside corner positions:
Symbolic inside positions:
Symbolic inside corner positions:
Use explicit coordinates to place the center of labels:
Place all labels at the upper-right corner of the vertex and vary the coordinates within the label:
Place multiple labels:
Any number of labels can be used:
Use the argument to Placed to control formatting including Tooltip:
Use more elaborate formatting functions:
Use any Graphics, Image, or Graphics3D as a vertex shape:
Specify vertex shapes for individual vertices:
VertexShape can be combined with VertexSize:
VertexShape is not affected by VertexStyle:
VertexShapeFunction has higher priority than VertexShape:
Get a list of built-in collections for VertexShapeFunction:
Use built-in settings for VertexShapeFunction in the collection:
Simple basic shapes:
Common basic shapes:
Use built-in settings for VertexShapeFunction in the collection:
Use built-in settings for VertexShapeFunction in the collection:
Draw individual vertices:
Combine with a default vertex function:
Draw vertices using a predefined graphic:
Draw vertices by running a program:
VertexShapeFunction can be combined with VertexStyle:
VertexShapeFunction has higher priority than VertexStyle:
VertexShapeFunction can be combined with VertexSize:
VertexShapeFunction has higher priority than VertexShape:
By default the size of vertices is computed automatically:
Specify the size of all vertices using symbolic vertex size:
Use a fraction of the minimum distance between vertex coordinates:
Use a fraction of the overall diagonal for all vertex coordinates:
Specify size in both the and directions:
Specify the size for individual vertices:
VertexSize can be combined with VertexShapeFunction:
VertexSize can be combined with VertexShape:
Style all vertices:
Style individual vertices:
VertexShapeFunction can be combined with VertexStyle:
VertexShapeFunction has higher priority than VertexStyle:
VertexStyle can be combined with BaseStyle:
VertexStyle has higher priority than BaseStyle:
VertexShape is not affected by VertexStyle:
 Applications   (7)
The GraphCenter of a complete graph includes all its vertices:
The GraphPeriphery includes all vertices:
The VertexEccentricity for all vertices is 1:
Highlight the vertex eccentricity path:
The GraphDiameter is 1:
Highlight the diameter path:
Vertex connectivity from to is the number of vertex-independent paths from to :
There are 3 vertex-independent paths between any pair of vertices:
The vertex connectivity for CompleteGraph[n] is :
Highlight the vertex degree for CompleteGraph:
Highlight the closeness centrality:
Highlight the eigenvector centrality:
Number of vertices of CompleteGraph[n]:
Number of edges of CompleteGraph[n]:
A complete graph is an -regular graph:
The subgraph of a complete graph is a complete graph:
The neighborhood of a vertex in a complete graph is the graph itself:
Complete graphs are their own cliques:
The GraphComplement of a complete graph with no edges:
For a complete graph, all entries outside the diagonal are 1s in the AdjacencyMatrix:
For a complete -partite graph, all entries outside the block diagonal are 1s:
The complete graph is the cycle graph :
The complete graph is the wheel graph :
The complete graph is the line graph of the star graph :
Random collage of complete graphs:
Coloring cycle decompositions in complete graphs on a prime number of vertices:
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