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Graph—Wolfram Language Documentation

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• See Also
  • TreeGraph
  • PathGraph
  • AdjacencyGraph
  • IncidenceGraph
  • GraphData
  • Graph3D
  • Entity Types
  • Graph
  • Interpreter Types
  • Graph
  • ComputedGraph
  • Formats
  • Graphlet
  • DOT
  • GraphML
• Related Guides
  • Graph Construction & Representation
  • Graph Visualization
  • Graphs & Networks
  • Discrete Mathematics
  • Social Network Analysis
  • Graph Layouts
  • WDF (Wolfram Data Framework)
  • Life Sciences & Medicine: Data & Computation
  • Scientific Models
  • Molecular Structure & Computation
  • Scientific Data Analysis
  • Database Connectivity
  • Wolfram Data Repository

Graph[{e₁,e₂,…}]

yields a graph with edges e_j.

Graph[{v₁,v₂,…},{e₁,e₂,…}]

yields the graph with vertices v_i and edges e_j.

Graph[{…,w_i[v_i,…],…},{…,w_j[e_j,…],…}]

yields a graph with vertex and edge properties defined by the symbolic wrappers w_k.

Graph[data]

yields a graph from data.

• Graph[…] displays in a notebook as a plot of a graph.
• Graph[…] is always converted to an optimized standard form with structure Graph[vertices,edges,…].
• Graph is treated as a raw object by functions like AtomQ, and for purposes of pattern matching.
• An undirected edge between u and v can be given as uv, u<->v, UndirectedEdge[u,v] or TwoWayRule[u,v]. The character  can be entered as ue.
• A directed edge from u to v can be given as uv, u->v, DirectedEdge[u,v], or Rule[u,v]. The character  can be entered as de.
• A tagged edge from u to v can be given as uv, uv, UndirectedEdge[u,v,t] or DirectedEdge[u,v,t].
• An undirected graph is specified using a collection of undirected edges.
• A directed graph is specified using a collection of directed edges.
• A mixed graph is specified using a collection of directed and undirected edges.
• The following special wrappers can be used for vertices and edges:
• Annotation[a,label] provide an annotation Button[a,action] define an action to execute when the element is clicked EventHandler[a,…] define a general event handler for the element Hyperlink[a,uri] make the element act as a hyperlink Labeled[a,…] display the element with labeling PopupWindow[a,cont] attach a popup window to the element StatusArea[a,label] display in the status area when the element is moused over Style[a,opts] show the element using the specified styles Tooltip[a,label] attach an arbitrary tooltip to the element
• The possible label placements are given in VertexLabels and EdgeLabels, respectively.
• Annotation can be used to associate annotations with vertices and edges:
• Annotation[v,name->value] associate the annotation name->value with the vertex v Annotation[e,name->value] associate the annotation name->value with the edge e
• The data can be any of the following:
• The following standard properties are supported for vertices:
• The following standard properties are supported for edges:
• Graph has the same options as Graphics, with the following additions and changes: [List of all options]
• Possible settings for PlotTheme include common base themes, color feature themes, font features themes, and size features themes.
• Graph feature themes affect the plots of vertices and edges. Feature themes include:
• AnnotationRules->{a->{name₁->val₁,…},…} associates the property name₁->val₁ etc. with a that can be a vertex, edge, or the graph itself. The following strings for a have special meanings:
• "DefaultEdgeProperties" default edge properties "DefaultVertexProperties" default vertex properties "GraphProperties" properties for the graph itself
• With the setting VertexCoordinates->Automatic, the placement of vertices and routing of edges is computed automatically, based on the setting for GraphLayout.
• Style and other specifications for edges are effectively applied in the order PlotTheme, EdgeStyle, GraphHighlightStyle, Style and other wrappers, and EdgeShapeFunction, with later specifications overriding earlier ones.
• Style and other specifications for vertices are effectively applied in the order PlotTheme, VertexStyle, GraphHighlightStyle, Style and other wrappers, and VertexShapeFunction, with later specifications overriding earlier ones.
• Label style and other specifications for edge labels are effectively applied in the order PlotTheme, EdgeLabelStyle, GraphHighlightStyle, Labeled, and EdgeLabels, with later specifications overriding earlier ones.
• Label style and other specifications for vertex labels are effectively applied in the order PlotTheme, VertexLabelStyle, GraphHighlightStyle, Labeled, and VertexLabels, with later specifications overriding earlier ones.

An undirected graph:

A directed graph:

Style vertices and edges:

Use wrappers to input directly:

Label vertices and edges:

Use different vertices and edges:

Create an undirected graph using  characters. Enter the character as ue:

Create a directed graph using  characters. Enter the character as de:

Create a directed graph from a list of rules:

Create an undirected graph from a list of rules:

Specify a graph with isolated vertices by giving an explicit list of vertices:

Use VertexList and EdgeList to get vertices and edges:

The ordering for edges is the order in which they were entered:

The ordering for vertices is the order in which they were entered in the edges:

Use an explicit vertex list to control the ordering used by VertexList:

The input vertex list controls the resulting vertex order:

Create undirected or directed graphs with self-loops:

Any expression can be used as vertices:

Use wrappers on vertices or edges:

Wrappers can be nested:

Add interactive behavior by wrappers such as Tooltip:

Any object can be used in the tooltip:

Use Button to trigger actions when clicking an edge or vertex:

Use PopupWindow to provide information drilldown:

Set the style for all vertices or edges:

Style individual vertices or edges using options:

Use wrappers for individual styling:

Adjust the size of vertices using symbolic sizes:

Or use sizes in terms of the smallest distance between vertex centers:

Use built-in collections for VertexShapeFunction:

Rounded shapes:

Concave shapes:

Draw individual vertices:

Combine with a default vertex function:

Use any Graphics, Image, or Graphics3D as a vertex shape:

Use built-in collections for EdgeShapeFunction:

Directed edges including solid arrows:

Line arrows:

Open arrows:

Specify an edge function for an individual edge:

Combine with a different default edge function:

Label any edge or vertex:

Use any expression as a label:

Control the placement of vertex labels using Placed, including symbolic inside positions:

Symbolic outside positions:

Coordinate-based positions:

Place multiple labels using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using VertexLabels:

Use Placed with symbolic locations to control label placement along an edge:

Use explicit coordinates to place labels:

Place multiple labels using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using EdgeLabels:

Specify an annotation for vertices:

Edges:

By default, a directed graph is generated when giving a list of rules:

Use DirectedEdges->False to interpret rules as undirected edges:

Use DirectedEdge or UndirectedEdge to directly specify whether a graph is directed or not:

Label the edge 12:

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 using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using EdgeLabels:

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:

Specify a weight for all edges:

Use any numeric expression as a weight:

Highlight the vertex 1:

Highlight the edge 23:

Highlight vertices and edges:

Use a common base theme:

Use a monochrome theme:

Use a large graph theme:

Use a classic diagram theme:

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 all labels at the center of vertices:

Place multiple labels using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using VertexLabels:

Use the argument to Placed to control formatting including Tooltip:

Or StatusArea:

Use more elaborate formatting functions:

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:

Set the weight for all vertices:

Use any numeric expression as a weight:

Build a graph of a finite mapping:

Draw the graph of a random permutation:

Use Table to set similar annotations for many items:

Generate a network of "nearby" words in a dictionary:

Graph objects are atomic raw objects:

Use GraphQ to test whether it is a graph:

Parallel edges are undistinguishable in Graph:

Use EdgeTaggedGraph to assign a unique tag to each edge:

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