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Emily Kirkman is working on this project. [http://sage.math.washington.edu:9001/graph Back to main wiki.] The Cayley graph for $A_5$:

{{{
sage: G = sage.groups.perm_gps.permgroup.AlternatingGroup(5)
sage: C = G.cayley_graph()
sage: C.show3d(bgcolor=(0,0,0), arc_color=(1,1,1), vertex_size=0.02, arc_size=0.007, arc_size2=0.01, xres=1000, yres=800, iterations=200)
}}}

{{attachment:A5.jpg}}

Emily Kirkman and Robert Miller are working on this project. [[http://wiki.sagemath.org/graph|Back to main wiki.]]
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Due to the volume of graphs now in the generators class, this wiki page is now intended to give status updates and serve as a gallery of graphs currently implemented. To see information on a specific graph, run SAGE or the SAGE [http://sage.math.washington.edu:8100 notebook]. For a list of graph constructors, type "graphs." and hit tab. For docstrings, type the graph name and one question mark (i.e.: "graphs.!CubeGraph?") then shift + enter. For source code, do likewise with two question marks.

[[TableOfContents]]		 Due to the volume of graphs now in the generators class, this wiki page is now intended to give status updates and serve as a gallery of graphs currently implemented. To see information on a specific graph, run SAGE or the SAGE [[http://sage.math.washington.edu:8100|notebook]]. For a list of graph constructors, type "graphs." and hit tab. For docstrings, type the graph name and one question mark (i.e.: "graphs.!CubeGraph?") then shift + enter. For source code, do likewise with two question marks.

<<TableOfContents>>
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}}}
attachment:chvatal.png		 sage: (graphs.ChvatalGraph()).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:chvatal.png}}
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}}}
attachment:desargues.png		 sage: (graphs.DesarguesGraph()).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:desargues.png}}
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attachment:flower.png {{attachment:flower.png}}
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attachment:frucht.png {{attachment:frucht.png}}
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attachment:heawood.png {{attachment:heawood.png}}
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attachment:moebiuskantor.png {{attachment:moebiuskantor.png}}
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}}}
attachment:pappus.png		 sage: (graphs.PappusGraph()).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:pappus.png}}
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attachment:petersen.png {{attachment:petersen.png}}
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attachment:thomsen.png {{attachment:thomsen.png}}
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attachment:compbip.png {{attachment:compbip.png}}
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attachment:complete.png {{attachment:complete.png}}
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attachment:cube.png {{attachment:cube.png}}
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attachment:biggercube.png {{attachment:biggercube.png}}
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}}}
attachment:baltree.png		 sage: (graphs.BalancedTree(3,5)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:baltree.png}}
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}}}
attachment:lcf.png		 sage: (graphs.LCFGraph(20, [-10,-7,-5,4,7,-10,-7,-4,5,7,-10,-7,6,-5,7,-10,-7,5,-6,7], 1)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:lcf.png}}
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attachment:tetrahedral.png {{attachment:tetrahedral.png}}
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}}}
attachment:hexahedral.png		 sage: (graphs.HexahedralGraph()).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:hexahedral.png}}
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attachment:octahedral.png {{attachment:octahedral.png}}
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}}}
attachment:icosahedral.png		 sage: (graphs.IcosahedralGraph()).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:icosahedral.png}}
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attachment:dodecahedral.png {{attachment:dodecahedral.png}}
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}}}
attachment:dorogovtsev.png		 sage: (graphs.DorogovtsevGoltsevMendesGraph(5)).show(figsize=[4,4], graph_border=True, vertex_size=10, vertex_labels=False)
}}}
{{attachment:tmp_6.png}}
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attachment:barbell.png {{attachment:barbell.png}}
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attachment:bull.png {{attachment:bull.png}}
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attachment:circladder.png {{attachment:circladder.png}}
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attachment:claw.png {{attachment:claw.png}}
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attachment:cycle.png {{attachment:cycle.png}}
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attachment:diamond.png {{attachment:diamond.png}}
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attachment:empty.png {{attachment:empty.png}}
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attachment:grid.png {{attachment:grid.png}}
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attachment:house.png {{attachment:house.png}}
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attachment:housex.png {{attachment:housex.png}}
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attachment:krack.png {{attachment:krack.png}}
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attachment:ladder.png {{attachment:ladder.png}}
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attachment:lollipop.png {{attachment:lollipop.png}}
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attachment:path.png {{attachment:path.png}}
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attachment:star.png {{attachment:star.png}}
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attachment:wheel.png {{attachment:wheel.png}}
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attachment:random.png {{attachment:random.png}}
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attachment:randomfast.png {{attachment:randomfast.png}}
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}}}
attachment:barabasi.png		 sage: (graphs.RandomBarabasiAlbert(7,3)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:barabasi.png}}
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}}}
attachment:gnm.png		 sage: (graphs.RandomGNM(7,16)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:gnm.png}}
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}}}
attachment:newman.png		 sage: (graphs.RandomNewmanWattsStrogatz(7,3,.5)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:newman.png}}
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}}}
attachment:holme.png		 sage: (graphs.RandomHolmeKim(12,3,.4)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:holme.png}}
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}}}
attachment:lobster.png		 sage: (graphs.RandomHolmeKim(12,3,.4)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:lobster.png}}
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}}}
attachment:powerlaw.png		 sage: (graphs.RandomTreePowerlaw(15)).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:powerlaw.png}}
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}}}
attachment:randreg.png		 sage: (graphs.RandomRegular(3,20)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:randreg.png}}
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}}}
attachment:shell.png		 sage: (graphs.RandomShell([(10,20,0.8),(20,40,0.8)])).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:shell.png}}
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{{{

}}}
attachment:randdir.png
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}}}
attachment:randdirgn.png		 sage: (graphs.RandomDirectedGN(12)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:randdirgn.png}}
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}}}
attachment:randdirgnc.png		 sage: (graphs.RandomDirectedGNC(12)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:randdirgnc.png}}
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}}}
attachment:randdirgnr.png		 sage: (graphs.RandomDirectedGNR(12,.15)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)
}}}
{{attachment:randdirgnr.png}}
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}}}
attachment:degseq.png		 sage: (graphs.DegreeSequence([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])).show(vertex_labels=False, node_size=30, figsize=[4,4], graph_border=True)
}}}
{{attachment:degseq.png}}
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}}}
attachment:degseqconf.png		 sage: (graphs.DegreeSequenceConfigurationModel([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])).show(vertex_labels=False, node_size=30, figsize=[4,4], graph_border=True)
}}}
{{attachment:degseqconf.png}}
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}}}
attachment:degseqtree.png		 sage: (graphs.DegreeSequenceTree([3,1,3,3,1,1,1,2,1])).show(figsize=[4,4], graph_border=True)
}}}
{{attachment:degseqtree.png}}
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}}}
attachment:degseqexp.png		 sage: (graphs.DegreeSequenceExpected([1,2,3,2,3])).show(figsize=[4,4],graph_border=True)
}}}
{{attachment:degseqexp.png}}

The Cayley graph for A_5: 

sage: G = sage.groups.perm_gps.permgroup.AlternatingGroup(5)
sage: C = G.cayley_graph()
sage: C.show3d(bgcolor=(0,0,0), arc_color=(1,1,1), vertex_size=0.02, arc_size=0.007, arc_size2=0.01, xres=1000, yres=800, iterations=200)

A5.jpg

Emily Kirkman and Robert Miller are working on this project. Back to main wiki.

The goal of the Graph Generators Class is to implement constructors for many common graphs, as well as thorough docstrings that can be used for reference. The graph generators will grow as the Graph Theory Project does. So please check back for additions and feel free to leave requests in the suggestions section.

We currently have 54 constructors of named graphs and basic structures. Most of these graphs are constructed with a preset dictionary of x-y coordinates of each node. This is advantageous for both style and time. (The default graph plotting in SAGE uses the spring-layout algorithm). SAGE graphs all have an associated graphics object, and examples of plotting options are shown on the graphs below.

As we implement algorithms into the Graph Theory Package, the constructors of known graphs would set their properties upon instantiation as well. For example, if someone created a very large complete bipartite graph and then asked if it is a bipartite graph (not currently implemented), then instead of running through an algorithm to check it, we could return a value set at instantiation. Further, this will improve the reference use of the docstrings as we would list the properties of each named graph.

Due to the volume of graphs now in the generators class, this wiki page is now intended to give status updates and serve as a gallery of graphs currently implemented. To see information on a specific graph, run SAGE or the SAGE notebook. For a list of graph constructors, type "graphs." and hit tab. For docstrings, type the graph name and one question mark (i.e.: "graphs.CubeGraph?") then shift + enter. For source code, do likewise with two question marks.

Contents

  1. Suggestions
  2. Graphs I Plan to Add
    1. Inherited from NetworkX
    2. Families of Graphs
    3. Named Graphs
  3. Gallery of Graph Generators in SAGE
    1. Named Graphs
      1. Chvatal Graph
      2. Desargues Graph
      3. Flower Snark
      4. Frucht
      5. Heawood
      6. Möbius Kantor
      7. Pappus Graph
      8. Petersen
      9. Thomsen
    2. Graph Families
      1. Complete Bipartite Graphs
      2. Complete Graphs
      3. Cube Graphs
      4. Balanced Tree
      5. LCF Graph
    3. Platonic Solids
      1. Tetrahedral Graph
      2. Hexahedral Graph
      3. Octahedral Graph
      4. Icosahedral Graph
      5. Dodecahedral Graph
    4. Pseudofractal Graphs
      1. Dorogovtsev Goltsev Mendes Graph
    5. Basic Structures
      1. Barbell Graph
      2. Bull Graph
      3. Circular Ladder Graph
      4. Claw Graph
      5. Cycle Graphs
      6. Diamond Graph
      7. Empty Graph
      8. Grid 2d Graph
      9. House Graph
      10. House X Graph
      11. Krackhardt Kite Graph
      12. Ladder Graph
      13. Lollipop Graph
      14. Path Graph
      15. Star Graph
      16. Wheel Graph
    6. Random Generators
      1. Random GNP
      2. Random GNP Fast
      3. Random Barabasi Albert
      4. Random GNM
      5. Random Newman Watts Strogatz
      6. Random Holme Kim
      7. Random Lobster
      8. Random Tree Powerlaw
      9. Random Regular
      10. Random Shell
    7. Random Directed Graphs
      1. Random Directed GN
      2. Random Directed GNC
      3. Random Directed GNR
    8. Graphs With a Given Degree Sequence
      1. Degree Sequence
      2. Degree Sequence Configuration Model
      3. Degree Sequence Tree
      4. Degree Sequence Expected

Suggestions

  • ???

Graphs I Plan to Add

Inherited from NetworkX

  • Bipartite Generators
  • Grid (n-dim)
  • Sedgewick
  • Truncated cube
  • Truncated tetrahedron
  • Tutte

Families of Graphs

  • Generalized Petersen graphs
  • Petersen Graph family
  • Trees (Directed – not simple. Maybe Balanced tree constructor and query isTree)
  • Cayley (Requires Edge Coloring)
  • Paley

Named Graphs

  • Brinkman
  • Clebsch
  • Grötzsch graph
  • Tutte eight-cage
  • Szekeres snark
  • Thomassen graph
  • Johnson (maybe own class)
  • Turan

Gallery of Graph Generators in SAGE

Named Graphs

Chvatal Graph

sage: (graphs.ChvatalGraph()).show(figsize=[4,4], graph_border=True)

chvatal.png

Desargues Graph

sage: (graphs.DesarguesGraph()).show(figsize=[4,4], graph_border=True)

desargues.png

Flower Snark

sage: flower_snark = graphs.FlowerSnark()
sage: flower_snark.set_boundary([15,16,17,18,19])
sage: flower_snark.show(figsize=[4,4], graph_border=True)

flower.png

Frucht

sage: frucht = graphs.FruchtGraph()
sage: frucht.show(figsize=[4,4], graph_border=True)

frucht.png

Heawood

sage: heawood = graphs.HeawoodGraph()
sage: heawood.show(figsize=[4,4], graph_border=True)

heawood.png

Möbius Kantor

sage: moebius_kantor = graphs.MoebiusKantorGraph()
sage: moebius_kantor.show(figsize=[4,4], graph_border=True)

moebiuskantor.png

Pappus Graph

sage: (graphs.PappusGraph()).show(figsize=[4,4], graph_border=True)

pappus.png

Petersen

sage: petersen = graphs.PetersenGraph()
sage: petersen.show(figsize=[4,4], graph_border=True)

petersen.png

Thomsen

sage: thomsen = graphs.ThomsenGraph()
sage: thomsen.show(figsize=[4,4], graph_border=True)

thomsen.png

Graph Families

Complete Bipartite Graphs

sage: comp_bip_list = []
sage: for i in range (2):
... for j in range (4):
...  comp_bip_list.append(graphs.CompleteBipartiteGraph(i+3,j+1))
...
sage: graphs_list.show_graphs(comp_bip_list)

compbip.png

Complete Graphs

sage: comp_list = []
sage: for i in range(13)[1:]:
... comp_list.append(graphs.CompleteGraph(i))
...
sage: graphs_list.show_graphs(comp_list)

complete.png

Cube Graphs

sage: cube_list = []
sage: for i in range(6)[2:]:
... cube_list.append(graphs.CubeGraph(i))
...
sage: graphs_list.show_graphs(cube_list)

cube.png 

sage: bigger_cube = graphs.CubeGraph(8)
sage: bigger_cube.show(figsize=[8,8], node_size=20, vertex_labels=False, graph_border=True)

biggercube.png

Balanced Tree

sage: (graphs.BalancedTree(3,5)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

baltree.png

LCF Graph

sage: (graphs.LCFGraph(20, [-10,-7,-5,4,7,-10,-7,-4,5,7,-10,-7,6,-5,7,-10,-7,5,-6,7], 1)).show(figsize=[4,4], graph_border=True)

lcf.png

Platonic Solids

Tetrahedral Graph

sage: tetrahedral = graphs.TetrahedralGraph()
sage: tetrahedral.show(figsize=[4,4], graph_border=True)

tetrahedral.png

Hexahedral Graph

sage: (graphs.HexahedralGraph()).show(figsize=[4,4], graph_border=True)

hexahedral.png

Octahedral Graph

sage: octahedral = graphs.OctahedralGraph()
sage: octahedral.show(figsize=[4,4], vertex_labels=False, node_size=50, graph_border=True)

octahedral.png

Icosahedral Graph

sage: (graphs.IcosahedralGraph()).show(figsize=[4,4], graph_border=True)

icosahedral.png

Dodecahedral Graph

sage: dodecahedral = graphs.DodecahedralGraph()
sage: dodecahedral.show(figsize=[4,4], vertex_labels=False, node_size=50, graph_border=True)

dodecahedral.png

Pseudofractal Graphs

Dorogovtsev Goltsev Mendes Graph

sage: (graphs.DorogovtsevGoltsevMendesGraph(5)).show(figsize=[4,4], graph_border=True, vertex_size=10, vertex_labels=False)

tmp_6.png

Basic Structures

Barbell Graph

sage: barbell_list = []
sage: for i in range (4):
... for j in range (2):
...  barbell_list.append(graphs.BarbellGraph(i+3, j+2))
...
sage: graphs_list.show_graphs(barbell_list)

barbell.png

Bull Graph

sage: bull = graphs.BullGraph()
sage: bull.show(figsize=[4,4], graph_border=True)

bull.png

Circular Ladder Graph

sage: circ_ladder = graphs.CircularLadderGraph(9)
sage: circ_ladder.show(figsize=[4,4], graph_border=True)

circladder.png

Claw Graph

sage: claw = graphs.ClawGraph()
sage: claw.show(figsize=[4,4], graph_border=True)

claw.png

Cycle Graphs

sage: cycle = graphs.CycleGraph(17)
sage: cycle.show(figsize=[4,4], graph_border=True)

cycle.png

Diamond Graph

sage: diamond = graphs.DiamondGraph()
sage: diamond.show(figsize=[4,4], graph_border=True)

diamond.png

Empty Graph

sage: empty = graphs.EmptyGraph()
sage: empty.show(figsize=[1,1], graph_border=True)

empty.png

Grid 2d Graph

sage: grid = graphs.Grid2dGraph(3,5)
sage: grid.show(figsize=[5,3])

grid.png

House Graph

sage: house = graphs.HouseGraph()
sage: house.show(figsize=[4,4], graph_border=True)

house.png

House X Graph

sage: houseX = graphs.HouseXGraph()
sage: houseX.show(figsize=[4,4], graph_border=True)

housex.png

Krackhardt Kite Graph

sage: krackhardt = graphs.KrackhardtKiteGraph()
sage: krackhardt.show(figsize=[4,4], graph_border=True)

krack.png

Ladder Graph

sage: ladder = graphs.LadderGraph(5)
sage: ladder.show(figsize=[4,4], graph_border=True)

ladder.png

Lollipop Graph

sage: lollipop_list = []
sage: for i in range (4):
... for j in range (2):
...  lollipop_list.append(graphs.LollipopGraph(i+3, j+2))
...
sage: graphs_list.show_graphs(lollipop_list)

lollipop.png

Path Graph

sage: path_line = graphs.PathGraph(5)
sage: path_circle = graphs.PathGraph(15)
sage: path_maze = graphs.PathGraph(45)
sage: path_list = [path_line, path_circle, path_maze]
sage: graphs_list.show_graphs(path_list)

path.png

Star Graph

sage: star_list = []
sage: for i in range (12)[4:]:
... star_list.append(graphs.StarGraph(i))
...
sage: graphs_list.show_graphs(star_list)

star.png

Wheel Graph

sage: wheel_list = []
sage: for i in range (12)[4:]:
... wheel_list.append(graphs.WheelGraph(i))
...
sage: graphs_list.show_graphs(wheel_list)

wheel.png

Random Generators

Random GNP

Use for dense graphs: 

time
sage: (graphs.RandomGNP(16,.77)).show(figsize=[4,4], graph_border=True)

My results: CPU time: 0.74 s, Wall time: 0.73 s random.png

Random GNP Fast

Use for sparse graphs: 

time
sage: (graphs.RandomGNPFast(16,.19)).show(figsize=[4,4], graph_border=True)

My results: CPU time: 0.63 s, Wall time: 0.62 s randomfast.png

Random Barabasi Albert

sage: (graphs.RandomBarabasiAlbert(7,3)).show(figsize=[4,4], graph_border=True)

barabasi.png

Random GNM

sage: (graphs.RandomGNM(7,16)).show(figsize=[4,4], graph_border=True)

gnm.png

Random Newman Watts Strogatz

sage: (graphs.RandomNewmanWattsStrogatz(7,3,.5)).show(figsize=[4,4], graph_border=True)

newman.png

Random Holme Kim

sage: (graphs.RandomHolmeKim(12,3,.4)).show(figsize=[4,4], graph_border=True)

holme.png

Random Lobster

sage: (graphs.RandomHolmeKim(12,3,.4)).show(figsize=[4,4], graph_border=True)

lobster.png

Random Tree Powerlaw

sage: (graphs.RandomTreePowerlaw(15)).show(figsize=[4,4], graph_border=True)

powerlaw.png

Random Regular

sage: (graphs.RandomRegular(3,20)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

randreg.png

Random Shell

sage: (graphs.RandomShell([(10,20,0.8),(20,40,0.8)])).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

shell.png

Random Directed Graphs

Random Directed GN

sage: (graphs.RandomDirectedGN(12)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

randdirgn.png

Random Directed GNC

sage: (graphs.RandomDirectedGNC(12)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

randdirgnc.png

Random Directed GNR

sage: (graphs.RandomDirectedGNR(12,.15)).show(node_size=20, vertex_labels=False, figsize=[4,4], graph_border=True)

randdirgnr.png

Graphs With a Given Degree Sequence

Degree Sequence

sage: (graphs.DegreeSequence([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])).show(vertex_labels=False, node_size=30, figsize=[4,4], graph_border=True)

degseq.png

Degree Sequence Configuration Model

sage: (graphs.DegreeSequenceConfigurationModel([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])).show(vertex_labels=False, node_size=30, figsize=[4,4], graph_border=True)

degseqconf.png

Degree Sequence Tree

sage: (graphs.DegreeSequenceTree([3,1,3,3,1,1,1,2,1])).show(figsize=[4,4], graph_border=True)

degseqtree.png

Degree Sequence Expected

sage: (graphs.DegreeSequenceExpected([1,2,3,2,3])).show(figsize=[4,4],graph_border=True)

degseqexp.png

graph_generators (last edited 2008-11-14 13:41:50 by anonymous)