({{, (RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph])[wmat] 
gives the graph with weighted adjacency matrix wmat.}, {, (RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph])[{(Subscript[v, 1]),(Subscript[v, 2]),…},wmat]
gives the graph with vertices (Subscript[v, i]) and weighted adjacency matrix wmat.}})

OpenersNotesSection(Spacer1)Details and Options

(RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph])[wmat] is equivalent to (RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph])[{1,2,…,n},wmat], where wmat has dimensions n×n.

(RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph]) takes the same options as (RefLink[Graph, paclet:ref/Graph]).

The option (RefLink[DirectedEdges, paclet:ref/DirectedEdges]) can be used to control whether an undirected or directed graph is constructed.

The following settings for (RefLink[DirectedEdges, paclet:ref/DirectedEdges]) can be used in (RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph]):

OpenersPrimaryExamplesSection(Spacer1)Examples  (95)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Construct a graph from a weighted adjacency matrix:

WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}}))]

A symmetric weighted adjacency matrix results in an undirected graph:

WeightedAdjacencyGraph[(({{∞, 1, 2}, {1, ∞, 1}, {2, 1, ∞}}))]

OpenersExampleSection1(Spacer1)Scope  (7)

Symmetric matrices are interpreted as undirected graphs:

WeightedAdjacencyGraph[(({{∞, 1, 2, ∞}, {1, ∞, 1, 1}, {2, 1, ∞, 3}, {∞, 1, 3, ∞}}))]

Unsymmetric matrices are interpreted as directed graphs:

WeightedAdjacencyGraph[(({{∞, 2, ∞, ∞}, {1, ∞, ∞, ∞}, {1, 1, ∞, ∞}, {3, 1, ∞, ∞}}))]

Use (RefLink[DirectedEdges, paclet:ref/DirectedEdges]) to control whether an undirected or directed graph should be constructed:

WeightedAdjacencyGraph[(({{∞, 1, 2, ∞}, {1, ∞, 1, 1}, {2, 1, ∞, 3}, {∞, 1, 3, ∞}})),DirectedEdges->True]

Matrices with diagonal entries create self-loops:

{WeightedAdjacencyGraph[(({{1, 1, 2, ∞}, {1, ∞, 1, 1}, {2, 1, ∞, 3}, {∞, 1, 3, ∞}}))],WeightedAdjacencyGraph[(({{∞, 2, ∞, ∞}, {1, ∞, ∞, ∞}, {1, 1, 1, ∞}, {3, 1, ∞, ∞}}))]}

Use a (RefLink[SparseArray, paclet:ref/SparseArray]) object to specify the adjacency matrix:

WeightedAdjacencyGraph[SparseArray[{{i_,j_}/;(i≠j)->1},{6,6},∞]]

By default, the vertices are taken to be the integers 1 through n:

VertexList@WeightedAdjacencyGraph[(({{∞, 2, 1, ∞}, {2, ∞, 1, 1}, {1, 1, ∞, 1}, {∞, 1, 1, ∞}}))]

Use an explicit vertex list to give vertex names:

VertexList@WeightedAdjacencyGraph[{a,b,c,d},(({{∞, 2, 1, ∞}, {2, ∞, 1, 1}, {1, 1, ∞, 1}, {∞, 1, 1, ∞}}))]

(RefLink[WeightedAdjacencyGraph, paclet:ref/WeightedAdjacencyGraph]) works with large matrices:

SparseArray[{Band[{2,1}]->1,Band[{3,1}]->1},{(10^6),(10^6)},∞]

Timing[WeightedAdjacencyGraph[%]//EdgeCount]

OpenersExampleSection2(Spacer1)Options  (82)

OpenersExampleSubsection0(Spacer1)DirectedEdges  (3)

By default, a symmetric matrix generates an undirected graph:

WeightedAdjacencyGraph[(({{∞, 2, 1, ∞}, {2, ∞, 1, 1}, {1, 1, ∞, 1}, {∞, 1, 1, ∞}}))]

Use (RefLink[DirectedEdges, paclet:ref/DirectedEdges])->(RefLink[True, paclet:ref/True]) to generate a directed graph:

WeightedAdjacencyGraph[(({{∞, 2, 1, ∞}, {2, ∞, 1, 1}, {1, 1, ∞, 1}, {∞, 1, 1, ∞}})),DirectedEdges->True,EdgeStyle->Arrowheads[Medium]]

By default, an unsymmetric matrix generates a directed graph:

WeightedAdjacencyGraph[(({{∞, 2, ∞, ∞}, {1, ∞, ∞, ∞}, {1, 1, ∞, ∞}, {3, 1, ∞, ∞}})),EdgeStyle->Arrowheads[Medium]]

OpenersExampleSubsection1(Spacer1)EdgeLabels  (7)

Label the edge 1<->2:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{1\[UndirectedEdge]2->"Hello"}]

Label all edges individually:

el={1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]1};

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->Table[el[[i]]->(Subscript["e", i]),{i,Length[el]}]]

Use any expression as a label:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{1\[UndirectedEdge]2->,2\[UndirectedEdge]3->,1\[UndirectedEdge]3-> }]

Use (RefLink[Placed, paclet:ref/Placed]) with symbolic locations to control label placement along an edge:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{1\[UndirectedEdge]2->Placed["■■■",p]},PlotLabel->p],{p,{"Start","Middle","End"}}]

Use explicit coordinates to place labels:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{1\[UndirectedEdge]2->Placed["■■■",p]},PlotLabel->p,BaselinePosition->Bottom],{p,{0,1/2,1}}]

Vary positions within the label:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{1\[UndirectedEdge]2->Placed["■■■",{1/2,p}]},PlotLabel->p,BaselinePosition->Bottom],{p,{{0,0},{1/2,1/2},{1,1}}}]

Place multiple labels using (RefLink[EdgeLabels, paclet:ref/EdgeLabels]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{3\[UndirectedEdge]1->Placed[{"lbl1","lbl2"},{"Start","End"}]}]

Any number of labels can be used:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->{3\[UndirectedEdge]1->Placed[{"lbl1","lbl2","lbl3"},{"Start","Middle","End"}]}]

Use automatic labeling by values through (RefLink[Tooltip, paclet:ref/Tooltip]) and (RefLink[StatusArea, paclet:ref/StatusArea]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->Placed["Name",Tooltip]]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeLabels->Placed["Name",StatusArea]]

OpenersExampleSubsection2(Spacer1)EdgeShapeFunction  (6)

Get a list of built-in settings for (RefLink[EdgeShapeFunction, paclet:ref/EdgeShapeFunction]):

GraphElementData["Edge"]

Undirected edges including the basic line:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeShapeFunction->"Line"]

Lines with different glyphs on the edges:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeShapeFunction->GraphElementData[ef,"ArrowSize"->0.1],PlotLabel->ef],{ef,{"BoxLine","DiamondLine","DotLine"}}]

Directed edges including solid arrows:

Table[WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}})),EdgeShapeFunction->GraphElementData[ef,"ArrowSize"->0.1],PlotLabel->ef],{ef,GraphElementData["Edge","FilledArrow"]}]

Line arrows:

Table[WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}})),EdgeShapeFunction->GraphElementData[ef,"ArrowSize"->0.1],PlotLabel->ef],{ef,GraphElementData["Edge","UnfilledArrow"]}]

Open arrows:

Table[WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}})),EdgeShapeFunction->GraphElementData[ef,"ArrowSize"->0.1],PlotLabel->ef],{ef,GraphElementData["Edge","CarvedArrow"]}]

Specify an edge function for an individual edge:

WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}})),EdgeShapeFunction->{1\[DirectedEdge]2->"FilledArrow"}]

Combine with a different default edge function:

WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 3}, {2, ∞, ∞}})),EdgeShapeFunction->{1\[DirectedEdge]2->"FilledArrow","CarvedArrow"}]

Draw edges by running a program:

ef[pts_List,e_]:=
Block[{s=0.015,g=},{Arrowheads[{{s,0.33,g},{s,0.67,g}}],Arrow[pts]}]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeShapeFunction->ef]

(RefLink[EdgeShapeFunction, paclet:ref/EdgeShapeFunction]) can be combined with (RefLink[EdgeStyle, paclet:ref/EdgeStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeStyle->Blue,EdgeShapeFunction->(Line[#1]&)]

(RefLink[EdgeShapeFunction, paclet:ref/EdgeShapeFunction]) has higher priority than (RefLink[EdgeStyle, paclet:ref/EdgeStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeStyle->Blue,EdgeShapeFunction->({Red,Line[#1]}&)]

OpenersExampleSubsection3(Spacer1)EdgeStyle  (2)

Style all edges:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeStyle->style,PlotLabel->style],{style,{Gray,Dashed,Thick}}]

Style individual edges:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeStyle->{1\[UndirectedEdge]2->Blue,1\[UndirectedEdge]3->Dashed}]

OpenersExampleSubsection4(Spacer1)EdgeWeight  (1)

Specify a weight for all edges:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),EdgeWeight->RandomInteger[5,3]]

WeightedAdjacencyMatrix[%]//MatrixForm

OpenersExampleSubsection5(Spacer1)GraphHighlight  (3)

Highlight the vertex 1:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->Tiny,GraphHighlight->{1}]

Highlight the edge 2<->3:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->Tiny,GraphHighlight->{2\[UndirectedEdge]3}]

Highlight vertices and edges:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->Tiny,GraphHighlight->{1,2,1\[UndirectedEdge]3,2\[UndirectedEdge]3}]

OpenersExampleSubsection6(Spacer1)GraphHighlightStyle  (2)

Get a list of built-in settings for (RefLink[GraphHighlightStyle, paclet:ref/GraphHighlightStyle]):

GraphElementData["GraphHighlightStyle"]

Use built-in settings for (RefLink[GraphHighlightStyle, paclet:ref/GraphHighlightStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),GraphHighlight->{1,2\[UndirectedEdge]3},VertexSize->Small,GraphHighlightStyle->#,PlotLabel->#]&/@Select[GraphElementData["GraphHighlightStyle"],#=!=Automatic&]

OpenersExampleSubsection7(Spacer1)GraphLayout  (5)

By default, the layout is chosen automatically:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),GraphLayout->Automatic]

Specify layouts on special curves:

m=WeightedAdjacencyMatrix[GridGraph[{10,10}]];

Table[WeightedAdjacencyGraph[m,0,GraphLayout->l,PlotLabel->l],{l,{"CircularEmbedding","SpiralEmbedding"}}]

Specify layouts that satisfy optimality criteria:

m=WeightedAdjacencyMatrix[GridGraph[{10,10}]];

Table[WeightedAdjacencyGraph[m,0,GraphLayout->l,PlotLabel->l],{l,{"SpringEmbedding","SpringElectricalEmbedding","HighDimensionalEmbedding"}}]

(RefLink[VertexCoordinates, paclet:ref/VertexCoordinates]) overrides (RefLink[GraphLayout, paclet:ref/GraphLayout]) coordinates:

{WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),GraphLayout->"SpringElectricalEmbedding"],
WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),GraphLayout->"SpringElectricalEmbedding",VertexCoordinates->Table[{i,i},{i,0,2}]]}

Use (RefLink[AbsoluteOptions, paclet:ref/AbsoluteOptions]) to extract (RefLink[VertexCoordinates, paclet:ref/VertexCoordinates]) computed using a layout algorithm:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}}))]

AbsoluteOptions[%,VertexCoordinates]

OpenersExampleSubsection8(Spacer1)PlotTheme  (4)

OpenersExampleSubsubsection0(Spacer1)Base Themes  (2)

Use a common base theme:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),PlotTheme->"Business"]

Use a monochrome theme:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),PlotTheme->"Monochrome"]

OpenersExampleSubsubsection1(Spacer1)Feature Themes  (2)

Use a large graph theme:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),PlotTheme->"LargeGraph"]

Use a classic diagram theme:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),PlotTheme->"ClassicDiagram"]

OpenersExampleSubsection9(Spacer1)Properties  (3)

Specify a property for vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),Properties->{1->{VertexLabels-> "hello"}}]

Edges:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),Properties->{1\[UndirectedEdge]2->{EdgeLabels-> "hello"}}]

Graph itself:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),Properties->{"GraphProperties"->{"Message"-> "hello"}}]

PropertyValue[%,"Message"]

OpenersExampleSubsection10(Spacer1)VertexCoordinates  (3)

By default, any vertex coordinates are computed automatically:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}}))]

Extract the resulting vertex coordinates using (RefLink[AbsoluteOptions, paclet:ref/AbsoluteOptions]):

AbsoluteOptions[%,VertexCoordinates]

Specify a layout function along an ellipse:

ellipseLayout[n_,{a_,b_}]:=Table[{a Cos[2Pi/n u],b Sin[2Pi/n u]},{u,1,n}]

Graphics[Point[ellipseLayout[20,{2,1}]]]

Use it to generate vertex coordinates for a graph:

m=WeightedAdjacencyMatrix[PathGraph[Range[20]]];

WeightedAdjacencyGraph[m,0,VertexCoordinates->ellipseLayout[20,{2,1}]]

(RefLink[VertexCoordinates, paclet:ref/VertexCoordinates]) has higher priority than (RefLink[GraphLayout, paclet:ref/GraphLayout]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexCoordinates->Table[{i,i},{i,3}],GraphLayout->"CircularEmbedding"]

OpenersExampleSubsection11(Spacer1)VertexLabels  (13)

Use vertex names as labels:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->"Name"]

Label individual vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->{1->"one"}]

Label all vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Table[i->(Subscript[v, i]),{i,3}]]

Use any expression as a label:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->{1->,2->,3->},ImagePadding->30]

Use (RefLink[Placed, paclet:ref/Placed]) with symbolic locations to control label placement, including outside positions:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.1,VertexShapeFunction->"Square",VertexLabels->Table[i->Placed["■■■",p],{i,3}],PlotLabel->p,ImagePadding->20],{p,{Before,After,Below,Above}}]

Symbolic outside corner positions:

pl={{Before,Below},{After,Below},{Before,Above},{After,Above}};

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.1,VertexShapeFunction->"Square",ImagePadding->20,VertexLabels->Table[i->Placed["■■■",p],{i,3}],PlotLabel->p],{p,pl}]

Symbolic inside positions:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.25,VertexLabels->Table[i->Placed["■■■",p],{i,3}],VertexShapeFunction->"Square",PlotLabel->p],{p,{Left,Top,Right,Bottom}}]

Symbolic inside corner positions:

pl={{Left,Bottom},{Right,Bottom},{Left,Top},{Right,Top}};

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.25,VertexShapeFunction->"Square",VertexLabels->Table[i->Placed["■■■",p],{i,3}],PlotLabel->p],{p,pl}]

Use explicit coordinates to place the center of labels:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.25,VertexShapeFunction->"Square",VertexLabels->Table[i->Placed[,p],{i,3}],PlotLabel->p,BaselinePosition->Bottom],{p,{{0,0},{1/2,1/2},{1,1}}}]

Place all labels at the upper-right corner of the vertex and vary the coordinates within the label:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.35,VertexShapeFunction->"Square",VertexLabels->Table[i->Placed[,{{1,1},p}],{i,3}],PlotLabel->p,BaselinePosition->Bottom],{p,{{0,0},{1/2,1/2},{1,1}}}]

Place multiple labels using (RefLink[Placed, paclet:ref/Placed]) in a wrapper:

WeightedAdjacencyGraph[{1,2,Labeled[3,Placed[{"lbl1","lbl2"},{Above,Below}]]},(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}}))]

Any number of labels can be used:

WeightedAdjacencyGraph[{1,2,Labeled[3,Placed[{"lbl1","lbl2","lbl3","lbl4"},{Above,After,Below,Before}]]},(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}}))]

Place multiple labels using (RefLink[VertexLabels, paclet:ref/VertexLabels]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->{3->Placed[{"lbl1","lbl2"},{Above,Below}]}]

Use the argument to (RefLink[Placed, paclet:ref/Placed]) to control formatting including (RefLink[Tooltip, paclet:ref/Tooltip]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Placed["Name",Tooltip]]

Or (RefLink[StatusArea, paclet:ref/StatusArea]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Placed["Name",StatusArea]]

Use more elaborate formatting functions:

rotateLabel[lbl_]:=Rotate[lbl,45Degree]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Table[i->Placed["xxx",Below,rotateLabel],{i,3}]]

panelLabel[lbl_]:=Panel[lbl,FrameMargins->0,Background->Lighter[Yellow,0.7]]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Table[i->Placed["xxx",Center,panelLabel],{i,3}]]

hyperlinkLabel[lbl_]:=Hyperlink[lbl,"http://www.wolfram.com"]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexLabels->Table[i->Placed["xxx",Center,hyperlinkLabel],{i,3}]]

OpenersExampleSubsection12(Spacer1)VertexShape  (5)

Use any (RefLink[Graphics, paclet:ref/Graphics]), (RefLink[Image, paclet:ref/Image]), or (RefLink[Graphics3D, paclet:ref/Graphics3D]) as a vertex shape:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShape->s,VertexSize->Medium],{s,{,,}}]

Specify vertex shapes for individual vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShape->{2->},VertexSize->Small]

(RefLink[VertexShape, paclet:ref/VertexShape]) can be combined with (RefLink[VertexSize, paclet:ref/VertexSize]):

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,VertexShape->,PlotLabel->s],{s,{Small,Large}}]

(RefLink[VertexShape, paclet:ref/VertexShape]) is not affected by (RefLink[VertexStyle, paclet:ref/VertexStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexShape->,VertexStyle->Blue]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) has higher priority than (RefLink[VertexShape, paclet:ref/VertexShape]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexShapeFunction->"Square",VertexShape->]

OpenersExampleSubsection13(Spacer1)VertexShapeFunction  (10)

Get a list of built-in collections for (RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]):

GraphElementData["Vertex"]

Use built-in settings for (RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) in the "Basic" collection:

GraphElementData["Vertex","Basic"]

Simple basic shapes:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->vf,VertexSize->0.2,PlotLabel->vf],{vf,{"Triangle","Square","Rectangle","Pentagon","Hexagon","Octagon"}}]

Common basic shapes:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->vf,VertexSize->0.2,PlotLabel->vf],{vf,{"DownTrapezoid","UpTrapezoid","Parallelogram","FiveDown","Circle","Diamond","Star","Capsule"}}]

Use built-in settings for (RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) in the "Rounded" collection:

GraphElementData["Vertex","Rounded"]

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->vf,VertexSize->0.2,PlotLabel->vf],{vf,GraphElementData["Vertex","Rounded"]}]

Use built-in settings for (RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) in the "Concave" collection:

GraphElementData["Vertex","Concave"]

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->vf,VertexSize->0.2,PlotLabel->vf],{vf,GraphElementData["Vertex","Concave"]}]

Draw individual vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->{ 1->"Square"},VertexSize->0.2]

Combine with a default vertex function:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->{ 1->"Square","Triangle"},VertexSize->0.2]

Draw vertices using a predefined graphic:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->(Inset[,#]&)]

Draw vertices by running a program:

vf[{xc_,yc_},name_,{w_,h_}]:=
Block[{xmin=xc-w,xmax=xc+w,ymin=yc-h,ymax=yc+h},
Polygon[{{xmin,ymin},{xmax,ymax},{xmin,ymax},{xmax,ymin}}]
];

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->vf,VertexSize->0.2]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) can be combined with (RefLink[VertexStyle, paclet:ref/VertexStyle]):

vf1[{xc_,yc_},name_,{w_,h_}]:=Rectangle[{xc-w,yc-h},{xc+w,yc+h}]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexStyle->Blue,VertexShapeFunction->vf1]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) has higher priority than (RefLink[VertexStyle, paclet:ref/VertexStyle]):

vf2[{xc_,yc_},name_,{w_,h_}]:={Red,Rectangle[{xc-w,yc-h},{xc+w,yc+h}]}

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexStyle->Blue,VertexShapeFunction->vf2]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) can be combined with (RefLink[VertexSize, paclet:ref/VertexSize]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexShapeFunction->"Star",VertexSize->{1->Small,Medium}]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) has higher priority than (RefLink[VertexShape, paclet:ref/VertexShape]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.3,VertexShapeFunction->"Star",VertexShape->]

OpenersExampleSubsection14(Spacer1)VertexSize  (8)

By default the size of vertices is computed automatically:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->Automatic]

Specify the size of all vertices using symbolic vertex size:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,PlotLabel->s],{s,{Tiny,Small,Medium,Large}}]

Use a fraction of the minimum distance between vertex coordinates:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,PlotLabel->s],{s,0.1,1,0.3}]

Use a fraction of the overall diagonal for all vertex coordinates:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->{"Scaled", s},PlotLabel->{"Scaled", s}],{s,0.1,1,0.3}]

Specify size in both the x and y directions:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,PlotLabel->s],{s,{{0.1,0.2},{0.2,0.1}}}]

Specify the size for individual vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->{1->0.2,2->0.3}]

(RefLink[VertexSize, paclet:ref/VertexSize]) can be combined with (RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]):

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,VertexShapeFunction->"Square",PlotLabel->s],{s,{0.05,0.1,0.2}}]

(RefLink[VertexSize, paclet:ref/VertexSize]) can be combined with (RefLink[VertexShape, paclet:ref/VertexShape]):

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->s,VertexShape->,PlotLabel->s],{s,{0.1,0.2,0.4}}]

OpenersExampleSubsection15(Spacer1)VertexStyle  (5)

Style all vertices:

Table[WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexStyle->style,VertexSize->0.2,PlotLabel->style],{style,{Yellow,EdgeForm[Dashed]}}]

Style individual vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexStyle->{1->Blue,2->Red},VertexSize->0.2]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) can be combined with (RefLink[VertexStyle, paclet:ref/VertexStyle]):

vf1[{xc_,yc_},name_,{w_,h_}]:=Rectangle[{xc-w,yc-h},{xc+w,yc+h}]

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexStyle->Blue,VertexShapeFunction->vf1]

(RefLink[VertexShapeFunction, paclet:ref/VertexShapeFunction]) has higher priority than (RefLink[VertexStyle, paclet:ref/VertexStyle]):

vf2[{xc_,yc_},name_,{w_,h_}]:={Red,Rectangle[{xc-w,yc-h},{xc+w,yc+h}]}

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexStyle->Blue,VertexShapeFunction->vf2]

(RefLink[VertexStyle, paclet:ref/VertexStyle]) can be combined with (RefLink[BaseStyle, paclet:ref/BaseStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexStyle->LightBlue,BaseStyle->EdgeForm[Dotted],VertexSize->0.2]

(RefLink[VertexStyle, paclet:ref/VertexStyle]) has higher priority than (RefLink[BaseStyle, paclet:ref/BaseStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexStyle->LightBlue,BaseStyle->Gray,VertexSize->0.2]

(RefLink[VertexShape, paclet:ref/VertexShape]) is not affected by (RefLink[VertexStyle, paclet:ref/VertexStyle]):

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexSize->0.2,VertexShape->,VertexStyle->Blue]

OpenersExampleSubsection16(Spacer1)VertexWeight  (2)

Set the weight for all vertices:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexWeight->{2,3,4}]

PropertyValue[{%,1},VertexWeight]

Use any numeric expression as a weight:

WeightedAdjacencyGraph[(({{∞, 1, 1}, {1, ∞, 1}, {1, 1, ∞}})),VertexWeight->{a,b,c}]

PropertyValue[{%,1},VertexWeight]

OpenersExampleSection3(Spacer1)Properties & Relations  (4)

Use (RefLink[VertexCount, paclet:ref/VertexCount]) and (RefLink[EdgeCount, paclet:ref/EdgeCount]) to count vertices and edges:

g=WeightedAdjacencyGraph[(({{∞, 1, 2, 1, 1}, {1, ∞, 1, 1, 1}, {2, 1, ∞, 1, 1}, {1, 1, 1, ∞, 3}, {1, 1, 1, 3, ∞}}))]

{VertexCount[g],EdgeCount[g]}

Use (RefLink[VertexList, paclet:ref/VertexList]) and (RefLink[EdgeList, paclet:ref/EdgeList]) to enumerate vertices and edges in standard order:

g=WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 1}, {1, ∞, ∞}}))]

{VertexList[g],EdgeList[g]}

Compute the (RefLink[WeightedAdjacencyMatrix, paclet:ref/WeightedAdjacencyMatrix]) from a graph:

g=WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 1}, {1, ∞, ∞}}))];

(m=WeightedAdjacencyMatrix[g])//MatrixForm

An adjacency matrix with all infinity entries in the diagonal constructs a graph without self-loops:

WeightedAdjacencyGraph[(({{∞, 1, ∞}, {∞, ∞, 1}, {1, ∞, ∞}}))]

LoopFreeGraphQ[%]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[WeightedAdjacencyMatrix, paclet:ref/WeightedAdjacencyMatrix])  ▪  (RefLink[WeightedGraphQ, paclet:ref/WeightedGraphQ])  ▪  (RefLink[EdgeWeightedGraphQ, paclet:ref/EdgeWeightedGraphQ])  ▪  (RefLink[VertexWeightedGraphQ, paclet:ref/VertexWeightedGraphQ])  ▪  (RefLink[EdgeWeight, paclet:ref/EdgeWeight])  ▪  (RefLink[VertexWeight, paclet:ref/VertexWeight])  ▪  (RefLink[AdjacencyGraph, paclet:ref/AdjacencyGraph])  ▪  (RefLink[IncidenceGraph, paclet:ref/IncidenceGraph])  ▪  (RefLink[KirchhoffGraph, paclet:ref/KirchhoffGraph])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Graphs and Matrices, paclet:guide/GraphsAndMatrices])}, {▪(RefLinkPlain[Graph Properties, paclet:guide/GraphProperties])}})}})