Opened 13 years ago

Closed 9 years ago

# more named graphs

Reported by: Owned by: mvngu jason, ncohen, rlm major sage-duplicate/invalid/wontfix graph theory brunellus N/A

### GitHub link to the corresponding issue

The database of common graphs currently implements lots of named graphs. Below is a list of named graphs to add to that database. See also ticket #2686:

### comment:1 Changed 13 years ago by ncohen

My god O_O SO you are basically saying I'm not sending enough ? :-D

To be honest I have tried to implement some of them, but felt I should ask for the help of Sage's algebraists... This one, for example : is there any way to build it using Sage's tools ?

Nathann

### comment:2 Changed 12 years ago by mvngu

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### comment:3 Changed 12 years ago by mvngu

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### comment:4 Changed 12 years ago by mvngu

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### comment:5 Changed 12 years ago by mvngu

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### comment:6 Changed 12 years ago by mvngu

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### comment:7 follow-ups:  8  17 Changed 12 years ago by dimpase

Many of these graphs can be trivially generated in GAP using its package Grape (a part of the optional gap_packages spkg) and a library of primitive groups (a part of optional databases_gap spkg). E.g. here is how to get Schlaefli graph:

```sage: gap.load_package('grape')
sage: gap.eval('G:=NullGraph(PrimitiveGroup(27,12),27);')
'rec( isGraph := true, order := 27, group := PSp(4, 3), \n  schreierVector := [ -1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, \n      2, 1, 1, 2, 2, 1, 1, 1, 2 ], adjacencies := [ [  ] ], \n  representatives := [ 1 ], isSimple := true )'
''
sage: gap.eval('VertexDegrees(G);')
'[ 10 ]'
sage: edges=gap('Orbit(G.group,[1,2],OnSets)')
sage: len(edges)
135
sage: schlaefli=Graph([[int(x[1])-1,int(x[2])-1] for x in edges])
sage: schlaefli.degree()
[10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10]
sage: schlaefli.diameter()
2
```

IMHO there is a more fundamental issue here: Sage should handle such graphs in an efficient way --- just keeping all the edges is pretty much a waste, in particular for bigger examples with hundreds of vertices...

### comment:8 in reply to:  7 Changed 12 years ago by dimpase

Many of these graphs can be trivially generated in GAP using its package Grape (a part of the optional gap_packages spkg) and a library of primitive groups (a part of optional databases_gap spkg).

actually, Grape isn't even needed (I mentioned it for illustrative purposes): to construct the Sage graph, all you need is the following:

```sage: edges=gap('Orbit(PrimitiveGroup(27,12),[1,2],OnSets)')
sage: schlaefli=Graph([[int(x[1])-1,int(x[2])-1] for x in edges])
```

PS. To get e.g. Hall-Janko graph, use `PrimitiveGroup(100,1)`...

### comment:9 Changed 12 years ago by mvngu

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### comment:10 Changed 12 years ago by mvngu

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### comment:11 Changed 12 years ago by mvngu

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### comment:12 Changed 12 years ago by mvngu

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### comment:13 Changed 12 years ago by mvngu

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### comment:14 follow-up:  40 Changed 12 years ago by dimpase

Fullerens are in fact a family, that can be generated. G.Brinkmann wrote a program called fullgen http://cs.anu.edu.au/~bdm/plantri/ that does just this, generating all non-isomorphic fullerens with given number of hexagonal faces. Unfortunately it has a weird license, so it cannot be just hooked up to Sage, at least not in a standard package.

### comment:15 follow-up:  16 Changed 12 years ago by ncohen

I have been sighing at plantri for a while.... I *need* to generate random planar graphs `:-p`

Nathann

### comment:16 in reply to:  15 Changed 12 years ago by dimpase

I have been sighing at plantri for a while.... I *need* to generate random planar graphs `:-p`

Sage way: throw random points on the sphere, generate the facets of their convex closuse (using e.g. cdd), then take the skeleton of the polytope (again, using cdd). Slow, but trivial to code :-)

### comment:17 in reply to:  7 ; follow-up:  18 Changed 12 years ago by rlm

IMHO there is a more fundamental issue here: Sage should handle such graphs in an efficient way --- just keeping all the edges is pretty much a waste, in particular for bigger examples with hundreds of vertices...

The underlying architecture is already in place; one needs only to implement a GraphBackend? which represents the graph in question. Implementing simple methods such as has_edge, has_vertex, etc. one can then get the rest of the methods automatically. Check out the source!

### comment:18 in reply to:  17 ; follow-up:  19 Changed 12 years ago by dimpase

IMHO there is a more fundamental issue here: Sage should handle such graphs in an efficient way --- just keeping all the edges is pretty much a waste, in particular for bigger examples with hundreds of vertices...

The underlying architecture is already in place; one needs only to implement a GraphBackend? which represents the graph in question. Implementing simple methods such as has_edge, has_vertex, etc. one can then get the rest of the methods automatically. Check out the source!

I am not sure I understand how to implement things like add_vertex() and add_edge() - as we would start with a permutation group G, the set of vertices is the domain of the group, and edges cannot be added one by one, but only as whole G-orbits. (Alternatively, not all orbits of G are used as the vertex set, and then adding a vertex would mean adding its G-orbit.)

### comment:19 in reply to:  18 Changed 12 years ago by rlm

I am not sure I understand how to implement things like add_vertex() and add_edge() - as we would start with a permutation group G, the set of vertices is the domain of the group, and edges cannot be added one by one, but only as whole G-orbits. (Alternatively, not all orbits of G are used as the vertex set, and then adding a vertex would mean adding its G-orbit.)

Well, you can always raise an error in the add_vertex function:

RuntimeError?: You can't add vertices to this kind of graph.

Or something similar. Then whenever you called a function which tried to add a vertex you would get that error, but the rest of the graph library would work just fine.

### comment:20 Changed 12 years ago by mvngu

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### comment:21 Changed 12 years ago by kini

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### comment:22 Changed 12 years ago by kini

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### comment:24 Changed 11 years ago by wdj

The following at 3-regular Hamiltonian graphs, hence covered by the LCF construction (http://en.wikipedia.org/wiki/LCF_notation):

```Balaban 10-cage
L = [-9, -25, -19, 29, 13, 35, -13, -29, 19, 25, 9, -29, 29, 17, 33, 21, 9,-13, -31, -9, 25, 17, 9, -31, 27, -9, 17, -19, -29, 27, -17, -9, -29, 33, -25,25, -21, 17, -17, 29, 35, -29, 17, -17, 21, -25, 25, -33, 29, 9, 17, -27, 29, 19, -17, 9, -27, 31, -9, -17, -25, 9, 31, 13, -9, -21, -33, -17, -29, 29]
G = graphs.LCFGraph(70, L, 1)

Balaban 11-cage
L = [44, 26, -47, -15, 35, -39, 11, -27, 38, -37, 43, 14, 28, 51, -29, -16, 41, -11, -26, 15, 22, -51, -35, 36, 52, -14, -33, -26, -46, 52, 26, 16, 43, 33, -15, 17, -53, 23, -42, -35, -28, 30, -22, 45, -44, 16, -38, -16, 50, -55, 20, 28, -17, -43, 47, 34, -26, -41, 11, -36, -23, -16, 41, 17, -51, 26, -33, 47, 17, -11, -20, -30, 21, 29, 36, -43, -52, 10, 39, -28, -17, -52, 51, 26, 37, -17, 10, -10, -45, -34, 17, -26, 27, -21, 46, 53, -10, 29, -50, 35, 15, -47, -29, -41, 26, 33, 55, -17, 42, -26, -36, 16]
G = graphs.LCFGraph(112, L, 1)

Bidiakis cube
G = graphs.LCFGraph(12, [6,4,-4], 4)

Biggs-Smith graph
L = [16, 24, -38, 17, 34, 48, -19, 41, -35, 47, -20, 34, -36, 21, 14, 48, -16, -36, -43, 28, -17, 21, 29, -43, 46, -24, 28, -38, -14, -50, -45, 21, 8, 27, -21, 20, -37, 39, -34, -44, -8, 38, -21, 25, 15, -34, 18, -28, -41, 36, 8, -29, -21, -48, -28, -20, -47, 14, -8, -15, -27, 38, 24, -48, -18, 25, 38, 31, -25, 24, -46, -14, 28, 11, 21, 35, -39, 43, 36, -38, 14, 50, 43, 36, -11, -36, -24, 45, 8, 19, -25, 38, 20, -24, -14, -21, -8, 44, -31, -38, -28, 37]
G = graphs.LCFGraph(102, L, 1)

Dyck graph
G = graphs.LCFGraph(32, [5,-5,13,-13], 8)

Foster graph
G = graphs.LCFGraph(90, [17,-9,37,-37,9,-17], 15)

Franklin graph
G = graphs.LCFGraph(12, [5,-5], 6)

Gray graph
G = graphs.LCFGraph(54, [-25,7,-7,13,-13,25], 9)

Harries graph
G = graphs.LCFGraph(70, [-29,-19,-13,13,21,-27,27,33,-13,13,19,-21,-33,29], 5)

Harries-Wong graph
L = [9, 25, 31, -17, 17, 33, 9, -29, -15, -9, 9, 25, -25, 29, 17, -9, 9, -27, 35, -9, 9, -17, 21, 27, -29, -9, -25, 13, 19, -9, -33, -17, 19, -31, 27, 11, -25, 29, -33, 13, -13, 21, -29, -21, 25, 9, -11, -19, 29, 9, -27, -19, -13, -35, -9, 9, 17, 25, -9, 9, 27, -27, -21, 15, -9, 29, -29, 33, -9, -25]
G = graphs.LCFGraph(70, L, 1)

Ljubljana graph
L = [47, -23, -31, 39, 25, -21, -31, -41, 25, 15, 29, -41, -19, 15, -49, 33, 39, -35, -21, 17, -33, 49, 41, 31, -15, -29, 41, 31, -15, -25, 21, 31, -51, -25, 23, 9, -17, 51, 35, -29, 21, -51, -39, 33, -9, -51, 51, -47, -33, 19, 51, -21, 29, 21, -31, -39]
G = graphs.LCFGraph(112, L, 2)

McGee graph
G = graphs.LCFGraph(24, [12,7,-7], 8)

Möbius–Kantor graph
G = graphs.LCFGraph(16, [5,-5], 8)

Nauru graph
G = graphs.LCFGraph(24, [5,-9,7,-7,9,-5], 4)

Tutte 12-cage
G = graphs.LCFGraph(126, [17, 27, -13, -59, -35, 35, -11, 13, -53, 53, -27, 21, 57, 11, -21, -57, 59, -17], 7)

Tutte–Coxeter graph
G = graphs.LCFGraph(30, [-13,-9,7,-7,9,13], 5)

Wagner graph
G = graphs.LCFGraph(8, [4], 8)
```

Some of the Fullerene graphs can be expressed in LCF notation as well.

### comment:25 Changed 11 years ago by kini

It's nice to have them explicitly constructed so you get a nice picture in a plot, though. I have a really old patch lying around for the Balaban 11-cage, I'll see if I can rebase it...

### comment:26 Changed 11 years ago by ncohen

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### comment:27 Changed 11 years ago by ncohen

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### comment:28 Changed 11 years ago by ncohen

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### comment:29 Changed 11 years ago by ncohen

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### comment:30 Changed 11 years ago by ncohen

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### comment:31 Changed 11 years ago by ncohen

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### comment:32 Changed 11 years ago by ncohen

Well David, all of your graphs are now Sage patches or are included already. My only regret is that I found not nice embedding for Tutte's 12 cage `:-/`

Nathann

### comment:33 Changed 11 years ago by ncohen

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### comment:34 Changed 11 years ago by ncohen

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### comment:35 Changed 11 years ago by ncohen

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### comment:36 Changed 11 years ago by chapoton

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### comment:37 Changed 10 years ago by chapoton

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### comment:38 Changed 10 years ago by chapoton

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### comment:39 Changed 10 years ago by chapoton

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### comment:40 in reply to:  14 ; follow-up:  41 Changed 10 years ago by nvcleemp

Fullerens are in fact a family, that can be generated. G.Brinkmann wrote a program called fullgen http://cs.anu.edu.au/~bdm/plantri/ that does just this, generating all non-isomorphic fullerens with given number of hexagonal faces. Unfortunately it has a weird license, so it cannot be just hooked up to Sage, at least not in a standard package.

Brinkmann's student J. Goedgebeur implemented a new version using a different algorithm which is faster for the `small' cases: http://caagt.ugent.be/buckygen/ This program is available under the GPL, so I assume it can be integrated in Sage. I'm willing to work on this. I have some familiarity with the program, since I integrated it into CaGe (http://caagt.ugent.be/CaGe).

### comment:41 in reply to:  40 Changed 10 years ago by ncohen

Brinkmann's student J. Goedgebeur implemented a new version using a different algorithm which is faster for the `small' cases: http://caagt.ugent.be/buckygen/ This program is available under the GPL, so I assume it can be integrated in Sage. I'm willing to work on this. I have some familiarity with the program, since I integrated it into CaGe (http://caagt.ugent.be/CaGe).

Wow ! Coooooooooool ! When you will create this ticket, could you please add in Cc : "ncohen,azi, Slani, Stefan" ? `:-)`

THaaaaaaaaaaanks !!

Nathann

Version 0, edited 10 years ago by ncohen (next)

### comment:42 Changed 10 years ago by nvcleemp

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### comment:43 Changed 10 years ago by chapoton

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### comment:44 Changed 10 years ago by chapoton

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### comment:45 Changed 9 years ago by ncohen

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### comment:46 Changed 9 years ago by ncohen

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### comment:47 Changed 9 years ago by ncohen

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### comment:48 Changed 9 years ago by ncohen

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### comment:49 Changed 9 years ago by ncohen

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### comment:50 Changed 9 years ago by chapoton

Maybe we can close that one ?

### comment:51 Changed 9 years ago by ncohen

Status: new → needs_review

I think that we can !

Funny.. It seemed unreachable when Minh created it... `;-)`

Nathann

### comment:52 Changed 9 years ago by ncohen

Milestone: sage-wishlist → sage-duplicate/invalid/wontfix needs_review → positive_review

Everything has been implemented.

### comment:53 Changed 9 years ago by vbraun

Resolution: → fixed positive_review → closed
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