Trac #30260: Graphs: more distance-regular graphs

Added a few more constructions of distance-regular graphs

URL: https://trac.sagemath.org/30260
Reported by: gh-Ivo-Maffei
Ticket author(s): Ivo Maffei
Reviewer(s): Dima Pasechnik

src/sage/graphs/generators/distance_regular.pyx

@@ -207,3 +207,172 @@ def graph_3O73():
     G = Graph(libgap.Orbit(group, [1, 3], libgap.OnSets), format='list_of_edges')
     G.name("Distance transitive graph with automorphism group 3.O_7(3)")
     return G
+
+def FosterGraph3S6():
+    r"""
+    Return the Foster graph for `3.Sym(6)`.
+
+    This graph is distance-regular with intersection array
+    `[6, 4, 2, 1; 1, 1, 4, 6]`.
+
+    The graph is also distance transitive.
+
+    EXAMPLES::
+
+        sage: G = graphs.FosterGraph3S6()
+        sage: G.is_distance_regular(True)
+        ([6, 4, 2, 1, None], [None, 1, 1, 4, 6])
+
+    REFERENCES:
+
+    A description and construction of this graph can be found in
+    [BCN1989]_ p. 397.
+    """
+
+    a = libgap.eval(("(2,6)(3,5)(4,11)(7,17)(8,16)(9,14)(13,22)(15,25)"
+                    "(18,29)(19,28)(20,21)(24,30)(26,35)(27,33)(31,39)"
+                     "(34,38)(36,43)(37,40)(42,44)"))
+    b = libgap.eval(("(1,2,7,12,4)(3,8,18,20,10)(5,9,19,21,11)(6,13,17,26,15)"
+                     "(14,23,28,31,24)(16,22,29,36,27)(25,32,35,42,34)"
+                     "(30,37,39,44,38)(33,40,43,45,41)"))
+
+    group = libgap.Group(a,b)
+
+    G = Graph(group.Orbit([1, 7], libgap.OnSets), format='list_of_edges')
+    G.name("Foster graph for 3.Sym(6) graph")
+    return G
+
+def J2Graph():
+    r"""
+    Return the distance-transitive graph with automorphism group `J_2`.
+
+    EXAMPLES::
+
+        sage: G = graphs.J2Graph()  # optional - internet gap_packages
+        sage: G.is_distance_regular(True) # optional - internet gap_packages
+        ([10, 8, 8, 2, None], [None, 1, 1, 4, 5])
+
+    REFERENCES:
+
+    A description and construction of this graph can be found in
+    [BCN1989]_ p. 408.
+    """
+    group = libgap.AtlasGroup("J2", libgap.NrMovedPoints, 315)
+    G = Graph(group.Orbit([1, 9], libgap.OnSets), format='list_of_edges')
+    G.name("J_2 graph")
+    return G
+
+def IvanovIvanovFaradjevGraph():
+    r"""
+    Return the IvanovIvanovFaradjev graph.
+
+    The graph is distance-transitive with automorphism group `3.M_{22}`.
+
+    EXAMPLES::
+
+        sage: G = graphs.IvanovIvanovFaradjevGraph()  # optional - internet gap_packages
+        sage: G.is_distance_regular(True)  # optional - internet gap_packages
+        ([7, 6, 4, 4, 4, 1, 1, 1, None], [None, 1, 1, 1, 2, 4, 4, 6, 7])
+
+    REFERENCES:
+
+    A description and construction of this graph can be found in
+    [BCN1989]_ p. 369.
+    """
+
+    group = libgap.AtlasGroup("3.M22", libgap.NrMovedPoints, 990)
+    graph = Graph(group.Orbit([1, 22], libgap.OnSets), format='list_of_edges')
+
+    graph.name("Ivanov-Ivanov-Faradjev Graph")
+    return graph
+
+def LargeWittGraph():
+    r"""
+    Return the large Witt graph.
+
+    This is a distance-regular graph with intersection array
+    `[30,28,24;1,3,15]`.
+
+    EXAMPLES::
+
+        sage: g = graphs.LargeWittGraph()
+        sage: g.is_distance_regular(True)
+        ([30, 28, 24, None], [None, 1, 3, 15])
+
+    REFERENCES:
+
+    A description of this graph can be found in
+    [BCN1989]_ p. 366.
+    This construction is taken from
+    http://mathworld.wolfram.com/LargeWittGraph.html
+    """
+    from sage.coding import codes_catalog as codes
+    import itertools
+
+    C = codes.GolayCode(GF(2), extended=True)
+    vertices = [c for c in C if c.hamming_weight() == 8]
+
+    edges = []
+    for v, w in itertools.combinations(vertices, 2):
+        if not set(v.support()).intersection(w.support()):
+            edges.append((v, w))
+
+    W = Graph(edges, format='list_of_edges')
+    W.name("Large Witt graph")
+    return W
+
+def TruncatedWittGraph():
+    r"""
+    Return the truncated Witt graph.
+
+    This builds the large Witt graph, then removes
+    all vertices whose codeword start with a 1.
+
+    The graph is distance-regular with intersection array
+    `[15,14,12;1,1,9]`.
+
+    EXAMPLES::
+
+         sage: G = graphs.TruncatedWittGraph()
+         sage: G.is_distance_regular(True)
+         ([15, 14, 12, None], [None, 1, 1, 9])
+
+    REFERENCES:
+
+    A description and construction of this graph can be found in
+    [BCN1989]_ p. 367.
+    """
+    # get large witt graph and remove all vertices which start with a 1
+    G = LargeWittGraph()
+    G.delete_vertices(filter(lambda x : x[0] == 1, G.vertices()))
+
+    G.name("Truncated Witt graph")
+    return G
+
+def DoublyTruncatedWittGraph():
+    r"""
+    Return the doubly truncated Witt graph.
+
+    This builds the truncated Witt graph, then removes
+    all vertices whose codeword start with a 1.
+
+    The graph is distance-regular with intersection array
+    `[7,6,4,4;1,1,1,6]`.
+
+    EXAMPLES::
+
+         sage: G = graphs.DoublyTruncatedWittGraph()
+         sage: G.is_distance_regular(True)
+         ([7, 6, 4, 4, None], [None, 1, 1, 1, 6])
+
+    REFERENCES:
+
+    A description and construction of this graph can be found in
+    [BCN1989]_ p. 368.
+    """
+
+    G = TruncatedWittGraph()
+    G.delete_vertices(filter(lambda x : x[1] == 1, G.vertices()))
+
+    G.name("Doubly Truncated Witt graph")
+    return G

src/sage/graphs/graph_generators.py

@@ -113,6 +113,7 @@ def __append_to_doc(methods):
      "DesarguesGraph",
      "DejterGraph",
      "DoubleStarSnark",
+     "DoublyTruncatedWittGraph",
      "DurerGraph",
      "DyckGraph",
      "EllinghamHorton54Graph",
@@ -122,6 +123,7 @@ def __append_to_doc(methods):
      "FlowerSnark",
      "FolkmanGraph",
      "FosterGraph",
+     "FosterGraph3S6",
      "FranklinGraph",
      "FruchtGraph",
      "GoldnerHararyGraph",
@@ -142,12 +144,15 @@ def __append_to_doc(methods):
      "HoltGraph",
      "HortonGraph",
      "IoninKharaghani765Graph",
+     "IvanovIvanovFaradjevGraph",
+     "J2Graph",
      "JankoKharaghaniGraph",
      "JankoKharaghaniTonchevGraph",
      "KittellGraph",
      "KrackhardtKiteGraph",
      "Klein3RegularGraph",
      "Klein7RegularGraph",
+     "LargeWittGraph",
      "locally_GQ42_distance_transitive_graph",
      "LocalMcLaughlinGraph",
      "LjubljanaGraph",
@@ -176,6 +181,7 @@ def __append_to_doc(methods):
      "TietzeGraph",
      "TruncatedIcosidodecahedralGraph",
      "TruncatedTetrahedralGraph",
+     "TruncatedWittGraph",
      "Tutte12Cage",
      "TutteCoxeterGraph",
      "TutteGraph",
@@ -1938,6 +1944,7 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
     DejterGraph              = staticmethod(smallgraphs.DejterGraph)
     DesarguesGraph           = staticmethod(smallgraphs.DesarguesGraph)
     DoubleStarSnark          = staticmethod(smallgraphs.DoubleStarSnark)
+    DoublyTruncatedWittGraph = staticmethod(distance_regular.DoublyTruncatedWittGraph)
     DurerGraph               = staticmethod(smallgraphs.DurerGraph)
     DyckGraph                = staticmethod(smallgraphs.DyckGraph)
     EllinghamHorton54Graph   = staticmethod(smallgraphs.EllinghamHorton54Graph)
@@ -1947,6 +1954,7 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
     FlowerSnark              = staticmethod(smallgraphs.FlowerSnark)
     FolkmanGraph             = staticmethod(smallgraphs.FolkmanGraph)
     FosterGraph              = staticmethod(smallgraphs.FosterGraph)
+    FosterGraph3S6           = staticmethod(distance_regular.FosterGraph3S6)
     FranklinGraph            = staticmethod(smallgraphs.FranklinGraph)
     FruchtGraph              = staticmethod(smallgraphs.FruchtGraph)
     GoldnerHararyGraph       = staticmethod(smallgraphs.GoldnerHararyGraph)
@@ -1968,12 +1976,15 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
     HoltGraph                = staticmethod(smallgraphs.HoltGraph)
     HortonGraph              = staticmethod(smallgraphs.HortonGraph)
     IoninKharaghani765Graph  = staticmethod(smallgraphs.IoninKharaghani765Graph)
+    IvanovIvanovFaradjevGraph = staticmethod(distance_regular.IvanovIvanovFaradjevGraph)
+    J2Graph                  = staticmethod(distance_regular.J2Graph)
     JankoKharaghaniGraph     = staticmethod(smallgraphs.JankoKharaghaniGraph)
     JankoKharaghaniTonchevGraph  = staticmethod(smallgraphs.JankoKharaghaniTonchevGraph)
     KittellGraph             = staticmethod(smallgraphs.KittellGraph)
     KrackhardtKiteGraph      = staticmethod(smallgraphs.KrackhardtKiteGraph)
     Klein3RegularGraph       = staticmethod(smallgraphs.Klein3RegularGraph)
     Klein7RegularGraph       = staticmethod(smallgraphs.Klein7RegularGraph)
+    LargeWittGraph           = staticmethod(distance_regular.LargeWittGraph)
     locally_GQ42_distance_transitive_graph = staticmethod(distance_regular.locally_GQ42_distance_transitive_graph)
     LocalMcLaughlinGraph     = staticmethod(smallgraphs.LocalMcLaughlinGraph)
     LjubljanaGraph           = staticmethod(smallgraphs.LjubljanaGraph)
@@ -2002,7 +2013,8 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
     TietzeGraph              = staticmethod(smallgraphs.TietzeGraph)
     Tutte12Cage              = staticmethod(smallgraphs.Tutte12Cage)
     TruncatedIcosidodecahedralGraph = staticmethod(smallgraphs.TruncatedIcosidodecahedralGraph)
-    TruncatedTetrahedralGraph= staticmethod(smallgraphs.TruncatedTetrahedralGraph)
+    TruncatedTetrahedralGraph = staticmethod(smallgraphs.TruncatedTetrahedralGraph)
+    TruncatedWittGraph       = staticmethod(distance_regular.TruncatedWittGraph)
     TutteCoxeterGraph        = staticmethod(smallgraphs.TutteCoxeterGraph)
     TutteGraph               = staticmethod(smallgraphs.TutteGraph)
     U42Graph216              = staticmethod(smallgraphs.U42Graph216)