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added a few sporadic distance regular graphs
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,212 @@ | ||
| r""" | ||
| Dabase of distance regular graphs | ||
| In this module we construct several distance regular graphs | ||
| and group them in a function that maps intersection arrays | ||
| to graphs. | ||
| For a survey on distance-regular graph see [BCN1989]_ or [VDKT2016]_. | ||
| EXAMPLES:: | ||
| sage: G = graphs.cocliques_HoffmannSingleton() | ||
| sage: G.is_distance_regular() | ||
| True | ||
| AUTHORS: | ||
| - Ivo Maffei (2020-07-28): initial version | ||
| """ | ||
|
|
||
| # **************************************************************************** | ||
| # Copyright (C) 2020 Ivo Maffei <ivomaffei@gmail.com> | ||
| # | ||
| # This program is free software: you can redistribute it and/or modify | ||
| # it under the terms of the GNU General Public License as published by | ||
| # the Free Software Foundation, either version 2 of the License, or | ||
| # (at your option) any later version. | ||
| # https://www.gnu.org/licenses/ | ||
| # **************************************************************************** | ||
|
|
||
| from sage.graphs.graph import Graph | ||
| from sage.libs.gap.libgap import libgap | ||
| from sage.modules.free_module import VectorSpace | ||
| from sage.modules.free_module_element import vector | ||
| from sage.rings.finite_rings.finite_field_constructor import GF | ||
|
|
||
| def cocliques_HoffmannSingleton(): | ||
| r""" | ||
| Return the graph obtained from the cocliques of the Hoffmann-Singleton graph. | ||
| This is a distance-regular graph with intersecion array | ||
| `[15, 14, 10, 3; 1, 5, 12, 15]`. | ||
| EXAMPLES:: | ||
| sage: G = graphs.cocliques_HoffmannSingleton() | ||
| sage: G.is_distance_regular(True) | ||
| ([15, 14, 10, 3, None], [None, 1, 5, 12, 15]) | ||
| REFERENCES: | ||
| The construction of this graph can be found in [BCN1989]_ p. 392. | ||
| """ | ||
| from sage.graphs.graph_generators import GraphGenerators | ||
| D = GraphGenerators.HoffmanSingletonGraph() | ||
| DC = D.complement() | ||
|
|
||
| cocliques = DC.cliques_maximum() # 100 of this | ||
|
|
||
| edges = [] | ||
| for i in range(100): | ||
| sC = frozenset(cocliques[i]) | ||
| for j in range(i+1,100): | ||
| if len(sC.intersection(cocliques[j])) == 8: | ||
| sC2 = frozenset(cocliques[j]) | ||
| edges.append( (sC,sC2) ) | ||
|
|
||
| G = Graph(edges,format="list_of_edges") | ||
| return G | ||
|
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||
|
|
||
| def locally_GQ42_graph(): | ||
| r""" | ||
| Return the unique amply regular graph which is locally a generalised | ||
| quadrangle. | ||
| This graph is distance-regular with intersection array | ||
| `[45, 32, 12, 1; 1, 6, 32, 45]`. | ||
| This graph is also distance-transitive. | ||
| EXAMPLES:: | ||
| sage: G = graphs.locally_GQ42_graph() | ||
| sage: G.is_distance_regular(True) | ||
| ([45, 32, 12, 1, None], [None, 1, 6, 32, 45]) | ||
| .. NOTE:: | ||
| This function needs the GAP's package AtlasRep [WPNBBAtl]_. | ||
| Install it via ``sage -i gap_packages``. | ||
| """ | ||
| H = libgap.AtlasGroup("3^2.U4(3).D8",libgap.NrMovedPoints,756) | ||
| Ns = H.NormalSubgroups() | ||
| for N in Ns: | ||
| if len(N.GeneratorsSmallest()) == 7: # there is only one | ||
| break | ||
|
|
||
| G = Graph(libgap.Orbit(N,[1,9],libgap.OnSets), format='list_of_edges') | ||
| G.name("locally GQ(4,2) graph") | ||
| return G | ||
|
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||
|
|
||
| def ConwaySmith_for_3S7(): | ||
| r""" | ||
| Return the Conway-Smith graph related to `3 Sym(7)`. | ||
| This is a distance-regular graph with intersection array | ||
| `[10, 6, 4, 1; 1, 2, 6, 10]`. | ||
| EXAMPLES:: | ||
| sage: G = graphs.ConwaySmith_for_3S7() | ||
| sage: G.is_distance_regular(True) | ||
| ([10, 6, 4, 1, None], [None, 1, 2, 6, 10]) | ||
| """ | ||
| from sage.rings.number_field.number_field import CyclotomicField | ||
|
|
||
| F = CyclotomicField(3) | ||
| w = F.gen() | ||
|
|
||
| V= VectorSpace(GF(4), 6) | ||
| z2 = GF(4)('z2') # GF(4) = {0,1,z2, z2+1} | ||
|
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||
| W = V.span([(0,0,1,1,1,1), (0,1,0,1,z2,z2+1), (1,0,0,1,z2+1,z2)]) | ||
| # we only need the 45 vectors with 2 zero entries | ||
| # we also embed everything into CC | ||
|
|
||
| K = [] | ||
| for v in W: | ||
| #check zero entries | ||
| zeros = 0 | ||
| for x in v: | ||
| if x == 0: | ||
| zeros += 1 | ||
|
|
||
| if zeros == 2: | ||
| # send to F and in K | ||
| # z2 -> w | ||
| # z2+1 -> w^2 | ||
| vv = [] # new vector | ||
| for x in v: | ||
| if x == z2: | ||
| vv.append(w) | ||
| elif x == z2+1: | ||
| vv.append(w**2) | ||
| else: | ||
| vv.append(int(x)) # this is weirdly needed for some reason | ||
|
|
||
| # now vv is the new vector in F | ||
| vv = vector(F, vv) | ||
| K.append(vv) | ||
|
|
||
| # we need to add other vectors | ||
| for i in range(6): | ||
| #create e_i | ||
| ei = [0]*6 | ||
| ei[i] = 1 | ||
| ei = vector(F, ei) | ||
|
|
||
| K.append(2 * ei) | ||
| K.append(2 * w * ei) | ||
| K.append(2 * w**2 * ei) | ||
| # now K is all the 63 vertices | ||
|
|
||
| def has_edge(u,v): | ||
| com = 0 | ||
| for i in range(6): | ||
| com += u[i].conjugate() * v[i] | ||
|
|
||
| if com == 2: | ||
| return True | ||
| return False | ||
|
|
||
| G = Graph() | ||
| length = len(K) | ||
| for i in range(length): | ||
| K[i].set_immutable() | ||
| for j in range(i+1, length): | ||
| if has_edge(K[i], K[j]): | ||
| K[j].set_immutable() | ||
| G.add_edge((K[i], K[j])) | ||
|
|
||
| G.name("Conway-Smith graph for 3S7") | ||
| return G | ||
|
|
||
| def graph_3O73(): | ||
| r""" | ||
| Return the graph related to the group `3 O(7,3)`. | ||
| This graph is distance-regular with intersection array | ||
| `[117, 80, 24, 1; 1, 12, 80, 117]`. | ||
| The graph is also distance transitive with "3.O(7,3)" as automorphism | ||
| group | ||
| EXAMPLES:: | ||
| sage: G = graphs.graph_3O73() | ||
| sage: G.is_distance_regular(True) | ||
| ([117, 80, 24, 1, None], [None, 1, 12, 80, 117]) | ||
| .. NOTE:: | ||
| This function needs the GAP's package AtlasRep [WPNBBAtl]_. | ||
| Install it via ``sage -i gap_packages``. | ||
| """ | ||
| group = libgap.AtlasGroup("3.O7(3)",libgap.NrMovedPoints,1134) | ||
| G = Graph(group.Orbit([1,3], libgap.OnSets), format='list_of_edges') | ||
| G.name("Distance transitive graph with automorphism group 3.O_7(3)") | ||
| return G |
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