# TorusGraph

TorusGraph[{n1,n2,,nk}]

gives the k-dimensional torus graph with n1×n2××nk vertices.

# Details and Options • TorusGraph is also known as toroidal graph.
• TorusGraph[{n1,n2,,nk}] gives a graph formed from the graph Cartesian product of the cycle graphs .
• • TorusGraph[{n1,n2,,nk},DirectedEdges->True] gives a directed torus graph.
• TorusGraph takes the same options as Graph.

# Examples

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## Basic Examples(2)

The first few torus graphs:

Higher-dimensional torus graphs:

Directed torus graphs:

## Scope(2)

Generate a k-dimensional torus graph:

1D:

2D:

3D: D:

Generate a directed torus graph:

## Options(83)

### AnnotationRules(3)

Specify an annotation for vertices:

Edges:

Graph itself:

### DirectedEdges(1)

By default, an undirected graph is generated:

Use to generate a directed graph:

### 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(3)

Specify a weight for all edges:

Use any numeric expression as a weight:

Specify weights for individual edges:

### 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(3)

Set the weight for all vertices:

Specify the weight for individual vertices:

Use any numeric expression as a weight:

## Applications(11)

### Basic Applications(6)

Visualize a torus graph:

Style vertices and edges of a torus graph:

Annotate vertices and edges of a torus graph:

Label a vertex:

Style an edge:

Modify a torus graph parameters:

Layout:

Generate a torus graph represented as a 3D plot:

Basic properties of the torus graph; the number of vertices:

The number of edges:

### Graph Theory(5)

Adjacency matrices are banded for a torus graph:

Assign distinct colors to adjacent vertices of a torus graph:

Visualize the graph:

Assign distinct colors to adjacent edges of a torus graph:

Visualize the graph:

Find the shortest tour in a simple torus graph:

Highlight the tour:

Find a spanning tree in a torus graph:

Highlight the tree:

## Properties & Relations(6)

TorusGraph[{n1,n2,}] has self-loops if one of is 1:

TorusGraph[{n1,n2,}] has parallel edges if one of is 2:

TorusGraph[{n}] is a cycle graph:

TorusGraph[{m,n}] is the graph formed from the Cartesian product of the cycle graphs and :

TorusGraph has the same number of vertices as GridGraph:

The difference between the number of edges of TorusGraph[f] and GridGraph[f]:

Same as:

## Possible Issues(1)

Large torus graphs do not automatically display as a plot of graph:

Use GraphPlot to plot the graph:

## Interactive Examples(1)

Animate by continuously changing the value of n: