CompleteGraph

CompleteGraph[n]

gives the complete graph with n vertices .

CompleteGraph[{n1,n2,,nk}]

gives the complete k-partite graph with n1+n2++nk vertices .

Details and Options

  • CompleteGraph generates a Graph object.
  • CompleteGraph[n] gives a graph with n vertices and an edge between every pair of vertices.
  • CompleteGraph[{n1,n2,,nk}] gives a graph with n1++nk vertices partitioned into disjoint sets Vi with ni vertices each and edges between all vertices in different sets Vi and Vj, but no edges between vertices in the same set Vi.
  • CompleteGraph[,DirectedEdges->True] gives a directed complete graph.
  • CompleteGraph takes the same options as Graph.

Examples

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Basic Examples  (4)

The first few complete graphs :

Bipartite graphs :

Directed complete graphs use two directional edges for each undirected edge:

Directed complete -partite graphs use directed edges from one group to another:

Options  (81)

AnnotationRules  (2)

Specify an annotation for vertices:

Edges:

DirectedEdges  (2)

By default, an undirected graph is generated:

Use DirectedEdges->True to generate a directed graph:

Generate directed -partite graphs:

EdgeLabels  (7)

Label the edge 12:

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:

EdgeShapeFunction  (6)

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:

EdgeStyle  (2)

Style all edges:

Style individual edges:

EdgeWeight  (2)

Specify a weight for all edges:

Use any numeric expression as a weight:

GraphHighlight  (3)

Highlight the vertex 1:

Highlight the edge 23:

Highlight the vertices and edges:

GraphHighlightStyle  (2)

Get a list of built-in settings for GraphHighlightStyle:

Use built-in settings for GraphHighlightStyle:

GraphLayout  (5)

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:

PlotTheme  (4)

Base Themes  (2)

Use a common base theme:

Use a monochrome theme:

Feature Themes  (2)

Use a large graph theme:

Use a classic diagram theme:

VertexCoordinates  (3)

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:

VertexLabels  (13)

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:

Or StatusArea:

Use more elaborate formatting functions:

VertexShape  (5)

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:

VertexShapeFunction  (10)

Get a list of built-in collections for VertexShapeFunction:

Use built-in settings for VertexShapeFunction in the "Basic" collection:

Simple basic shapes:

Common basic shapes:

Use built-in settings for VertexShapeFunction in the "Rounded" collection:

Use built-in settings for VertexShapeFunction in the "Concave" 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:

VertexSize  (8)

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:

VertexStyle  (5)

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:

VertexWeight  (2)

Set the weight for all vertices:

Use any numeric expression as a weight:

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 GraphRadius is 1:

Highlight the radius 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:

Properties & Relations  (12)

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 :

Neat Examples  (2)

Random collage of complete graphs:

Coloring cycle decompositions in complete graphs on a prime number of vertices:

Introduced in 2010
 (8.0)