more named graphs

[mvngu]

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:

1. ​Bidiakis cube --- #10307
2. ​Brinkmann graph --- #10310
3. ​Butterfly graph --- #10313
4. ​Friendship graph --- #10315
5. ​Dürer graph --- #10316
6. ​Errera graph --- #10321
7. ​Franklin graph --- #10322
8. ​Goldner–Harary graph --- #10329
9. ​Grötzsch graph --- #10330
10. ​Herschel graph --- #10337
11. ​Moser spindle --- #10338
12. ​Balaban 10-cage --- #12942
13. ​Balaban 11-cage --- #12945
14. ​Double-star snark --- #12952
15. ​Ellingham–Horton graph --- #12989
16. ​fullerene graphs --- #14618
17. ​Harries graph--- #12952
18. ​Harries–Wong graph --- #12980
19. ​Hoffman graph --- #13038
20. ​Holt graph --- #13411
21. ​Horton graph --- #15049
22. ​Kittell graph --- #15049
23. ​Markström graph --- #15049
24. ​McGee graph --- #12982
25. ​Meredith graph -- #15044
26. ​Sousselier graph --- #15049
27. ​Poussin graph --- #15049
28. ​Robertson graph --- #14911
29. ​Tutte's fragment
30. ​Tutte graph --- #15049
31. ​Young–Fibonacci lattice
32. ​Wagner graph --- #12982
33. ​Wiener-Araya graph -- #15049
34. ​Clebsch graph --- #13038
35. ​Hall–Janko graph --- #13058
36. ​Paley graph --- #13088
37. ​Shrikhande graph --- #10781
38. ​Möbius–Kantor graph (done)
39. ​Nauru graph --- #13862
40. ​Coxeter graph --- #13038
41. ​Tutte–Coxeter graph --- #12982
42. ​Dyck graph --- #10790
43. ​Foster graph --- #12952
44. ​Biggs–Smith graph --- #12971
45. ​Rado graph
46. ​Folkman graph --- #14904
47. ​Gray graph --- #12952
48. ​Ljubljana graph --- #12981
49. ​Tutte 12-cage --- #12982
50. ​Blanuša snarks -- #15054
51. ​Szekeres snark -- #15054
52. ​Tietze's graph -- #15054
53. ​Watkins snark -- #15054

[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 ?

​http://www.win.tue.nl/~aeb/graphs/Schlaefli.html

Nathann

[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('AddEdgeOrbit(G,[1,2]);')
''
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...

[dimpase]

Replying to 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)...

[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.

[ncohen]

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

Nathann

[dimpase]

Replying to ncohen:

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 :-)

[rlm]

Replying to 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!

[dimpase]

Replying to rlm:

Replying to 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.)

[rlm]

Replying to dimpase:

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.

[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.

[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...

[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

[nvcleemp]

Replying to 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.

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).

[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

[chapoton]

Maybe we can close that one ?

[ncohen]

I think that we can !

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

Nathann

[ncohen]

Everything has been implemented.