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Upgrading from:

DiscreteMath`GraphPlot`

GraphPlot functionality is now available in the newly added built-in Mathematica kernel functions GraphPlot and LayeredGraphPlot.
GraphPlot3D is now available as the newly added built-in Mathematica kernel function GraphPlot3D.
TreePlot is now available as the newly added built-in Mathematica kernel function TreePlot.
GraphDistance, PseudoDiameter, MaximalBipartiteMatching, MaximalIndependentVertexSet, MaximalIndependentEdgeSet, MinCut, StrongComponents, and VertexList are now available through the newly created Graph Utilities Package.
The EdgeStyleFunction option is now available as the newly added built-in Mathematica kernel option EdgeRenderingFunction.
The TreeSizeFunction option is now available as the newly added built-in Mathematica kernel option LayerSizeFunction.
The VertexCoordinates option is now available as the newly added built-in Mathematica kernel option VertexCoordinateRules.
The VertexStyleFunction option is now available as the newly added built-in Mathematica kernel option VertexRenderingFunction.
The RootPosition option is now available as the second argument to the built-in Mathematica kernel function TreePlot.

GraphPlot

Plot a graph specified by a rule list:

Version 5.2 << DiscreteMath`GraphPlot`; g = GraphPlot[{2 -> 1, 3 -> 1, 3 -> 2, 4 -> 1, 4 -> 2, 5 -> 1, 5 -> 2}]

GraphPlot3D

Lay out a graph in 3D:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot3D[{2 -> 1, 3 -> 1, 3 -> 2, 4 -> 1, 4 -> 2, 5 -> 1, 5 -> 2}]

TreePlot

Draw a binary tree:

Version 5.2 << DiscreteMath`GraphPlot`; g = Flatten[Table[{i -> 2*i, i -> 2*i + 1}, {i, 2^4 - 1}]] TreePlot[g];

EdgeStyleFunction

Use EdgeShapeFunction to control the way edges are drawn:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[Table[1, {30}, {30}], EdgeStyleFunction -> ({} &)]

Alternate ways to draw edges as arrows:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[{1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5, 5 -> 6, 6 -> 7, 7 -> 8, 8 -> 1, 1 -> 9, 2 -> 9, 3 -> 10, 4 -> 10, 6 -> 11, 5 -> 11, 7 -> 12, 8 -> 12}, EdgeStyleFunction -> (Arrow[{#1, #2}] &)];

LayerSizeFunction

Draw a binary tree:

Version 5.2 << DiscreteMath`GraphPlot`; g = Flatten[Table[{i -> 2*i, i -> 2*i + 1}, {i, 2^4 - 1}]] TreePlot[g];

This plots the same graph, but with the height of each level i proportional to 1/i2:

Version 5.2 << DiscreteMath`GraphPlot`; g = Flatten[Table[{i -> 2*i, i -> 2*i + 1}, {i, 2^4 - 1}]] TreePlot[g, TreeSizeFunction -> (1/#^2 &)];

VertexCoordinates

Use VertexCoordinates to specify explicit coordinates for vertices:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[{1 -> 3, 1 -> 4, 2 -> 4, 2 -> 5, 3 -> 5, 6 -> 7, 7 -> 8, 8 -> 9, 9 -> 10, 6 -> 10, 1 -> 6, 2 -> 7, 3 -> 8, 4 -> 9, 5 -> 10}, VertexCoordinates -> {{0.309017, 0.951057}, {-0.809017, -0.587785}, {0.309017, -0.951057}, \ {-0.809017, 0.587785}, {1., 0}, {0.618034, 1.90211}, {-1.61803, 1.17557}, {-1.61803, -1.17557}, {0.618034, -1.90211}, {2., 0}}

VertexStyleFunction

Use VertexShapeFunction to control the way vertices are drawn. This draws no vertices:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[{1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5, 5 -> 6, 6 -> 1}, VertexStyleFunction -> ({} &)]

This draws large blue vertices:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[{1 -> 2, 1 -> 3, 1 -> 4, 2 -> 3, 2 -> 4, 3 -> 5, 4 -> 6, 5 -> 7, 5 -> 8, 6 -> 7, 6 -> 8, 7 -> 8}, VertexStyleFunction -> ({Hue[0], Disk[#, 0.03]} &)];

RootPosition (Argument to TreePlot)

This plots the tree by placing the root to the left:

Version 5.2 << DiscreteMath`GraphPlot`; TreePlot[{1 -> 2, 2 -> 4, 3 -> 6, 4 -> 8, 5 -> 10, 6 -> 12, 1 -> 3, 2 -> 5, 3 -> 7, 4 -> 9, 5 -> 11, 6 -> 13}, RootPosition -> Left]
GraphPlot

Many functions from DiscreteMath`GraphPlot` are now available as built-in Mathematica kernel functions.

This plots a graph of 37 vertices:

Version 5.2 << DiscreteMath`GraphPlot`; GraphPlot[\!\(\* TagBox[ RowBox[{"g", "=", RowBox[{"{", RowBox[{ RowBox[{"1", "->", "2"}], ",", RowBox[{"1", "->", "6"}], ",", RowBox[{"1", "->", "7"}], ",", RowBox[{"2", "->", "3"}], ",", RowBox[{"2", "->", "7"}], ",", RowBox[{"3", "->", "30"}], ",", RowBox[{"3", "->", "26"}], ",", RowBox[{"3", "->", "4"}], ",", RowBox[{"3", "->", "32"}], ",", RowBox[{"4", "->", "5"}], ",", RowBox[{"4", "->", "9"}], ",", RowBox[{"5", "->", "9"}], ",", RowBox[{"5", "->", "10"}], ",", RowBox[{"6", "->", "7"}], ",", RowBox[{"6", "->", "11"}], ",", RowBox[{"7", "->", "8"}], ",", RowBox[{"7", "->", "30"}], ",", RowBox[{"7", "->", "12"}], ",", RowBox[{"8", "->", "9"}], ",", RowBox[{"8", "->", "26"}], ",", RowBox[{"8", "->", "13"}], ",", RowBox[{"9", "->", "10"}], ",", RowBox[{"9", "->", "32"}], ",", RowBox[{"9", "->", "14"}], ",", RowBox[{"10", "->", "15"}], ",", RowBox[{"11", "->", "16"}], ",", RowBox[{"11", "->", "27"}], ",", RowBox[{"11", "->", "30"}], ",", RowBox[{"11", "->", "35"}], ",", RowBox[{"12", "->", "13"}], ",", RowBox[{"12", "->", "17"}], ",", RowBox[{"12", "->", "27"}], ",", RowBox[{"13", "->", "31"}], ",", RowBox[{"13", "->", "33"}], ",", RowBox[{"13", "->", "14"}], ",", RowBox[{"13", "->", "36"}], ",", RowBox[{"13", "->", "18"}], ",", RowBox[{"13", "->", "34"}], ",", RowBox[{"14", "->", "28"}], ",", RowBox[{"14", "->", "19"}], ",", RowBox[{"15", "->", "32"}], ",", RowBox[{"15", "->", "28"}], ",", RowBox[{"15", "->", "37"}], ",", RowBox[{"15", "->", "20"}], ",", RowBox[{"16", "->", "17"}], ",", RowBox[{"16", "->", "21"}], ",", RowBox[{"17", "->", "21"}], ",", RowBox[{"17", "->", "22"}], ",", RowBox[{"17", "->", "35"}], ",", RowBox[{"17", "->", "18"}], ",", RowBox[{"18", "->", "19"}], ",", RowBox[{"18", "->", "29"}], ",", RowBox[{"19", "->", "37"}], ",", RowBox[{"19", "->", "20"}], ",", RowBox[{"19", "->", "24"}], ",", RowBox[{"19", "->", "25"}], ",", RowBox[{"20", "->", "25"}], ",", RowBox[{"21", "->", "22"}], ",", RowBox[{"22", "->", "23"}], ",", RowBox[{"23", "->", "24"}], ",", RowBox[{"23", "->", "29"}], ",", RowBox[{"23", "->", "35"}], ",", RowBox[{"23", "->", "37"}], ",", RowBox[{"24", "->", "25"}], ",", RowBox[{"30", "->", "31"}], ",", RowBox[{"32", "->", "33"}], ",", RowBox[{"34", "->", "35"}], ",", RowBox[{"36", "->", "37"}]}], "}"}]}], Short[#, 1000]& ]\), VertexStyleFunction -> Automatic];

This calculates the diameter of the graph:

Version 5.2 << DiscreteMath`GraphPlot`; ug = Join[g, Reverse /@ g]; PseudoDiameter[ug]

This calculates the distance between vertices 1 and 25:

Version 5.2 << DiscreteMath`GraphPlot`; GraphDistance[ug, 1, 25]