Python Graph.vertices方法代码示例

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import vertices [as 别名]
    def __classcall_private__(cls, G, names=None):
        """
        Normalize input to ensure a unique representation.

        TESTS::

            sage: G1 = RightAngledArtinGroup(graphs.CycleGraph(5))
            sage: Gamma = Graph([(0,1),(1,2),(2,3),(3,4),(4,0)])
            sage: G2 = RightAngledArtinGroup(Gamma)
            sage: G3 = RightAngledArtinGroup([(0,1),(1,2),(2,3),(3,4),(4,0)])
            sage: G4 = RightAngledArtinGroup(Gamma, 'v')
            sage: G1 is G2 and G2 is G3 and G3 is G4
            True

        Handle the empty graph::

            sage: RightAngledArtinGroup(Graph())
            Traceback (most recent call last):
            ...
            ValueError: the graph must not be empty
        """
        if not isinstance(G, Graph):
            G = Graph(G, immutable=True)
        else:
            G = G.copy(immutable=True)
        if G.num_verts() == 0:
            raise ValueError("the graph must not be empty")
        if names is None:
            names = 'v'
        if isinstance(names, six.string_types):
            if ',' in names:
                names = [x.strip() for x in names.split(',')]
            else:
                names = [names + str(v) for v in G.vertices()]
        names = tuple(names)
        if len(names) != G.num_verts():
            raise ValueError("the number of generators must match the"
                             " number of vertices of the defining graph")
        return super(RightAngledArtinGroup, cls).__classcall__(cls, G, names)

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import vertices [as 别名]
def _check_pbd(B,v,S):
    r"""
    Checks that ``B`` is a PBD on `v` points with given block sizes.

    INPUT:

    - ``bibd`` -- a list of blocks

    - ``v`` (integer) -- number of points

    - ``S`` -- list of integers

    EXAMPLE::

        sage: designs.BalancedIncompleteBlockDesign(40,4).blocks() # indirect doctest
        [[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
         [0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
         [0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
        ...
    """
    from itertools import combinations
    from sage.graphs.graph import Graph

    if not all(len(X) in S for X in B):
        raise RuntimeError("This is not a nice honest PBD from the good old days !")

    g = Graph()
    m = 0
    for X in B:
        g.add_edges(list(combinations(X,2)))
        if g.size() != m+binomial(len(X),2):
            raise RuntimeError("This is not a nice honest PBD from the good old days !")
        m = g.size()

    if not (g.is_clique() and g.vertices() == range(v)):
        raise RuntimeError("This is not a nice honest PBD from the good old days !")

    return B

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import vertices [as 别名]
    def graph(self):
        """
        Returns a graph whose vertices correspond to curves in this class, and whose edges correspond to prime degree isogenies.

        .. note:

            There are only finitely many possible isogeny graphs for
            curves over Q [M78].  This function tries to lay out the graph
            nicely by special casing each isogeny graph.

        .. note:

            The vertices are labeled 1 to n rather than 0 to n-1 to
            correspond to LMFDB and Cremona labels.

        EXAMPLES::

            sage: isocls = EllipticCurve('15a3').isogeny_class(use_tuple=False)
            sage: G = isocls.graph()
            sage: sorted(G._pos.items())
            [(1, [-0.8660254, 0.5]), (2, [-0.8660254, 1.5]), (3, [-1.7320508, 0]), (4, [0, 0]), (5, [0, -1]), (6, [0.8660254, 0.5]), (7, [0.8660254, 1.5]), (8, [1.7320508, 0])]

        REFERENCES:

        .. [M78] B. Mazur.  Rational Isogenies of Prime Degree.
          *Inventiones mathematicae* 44,129-162 (1978).
        """
        from sage.graphs.graph import Graph
        M = self.matrix(fill = False)
        n = M.nrows() # = M.ncols()
        G = Graph(M, format='weighted_adjacency_matrix')
        N = self.matrix(fill = True)
        D = dict([(v,self.curves[v]) for v in G.vertices()])
        # The maximum degree classifies the shape of the isogeny
        # graph, though the number of vertices is often enough.
        # This only holds over Q, so this code will need to change
        # once other isogeny classes are implemented.
        if n == 1:
            # one vertex
            pass
        elif n == 2:
            # one edge, two vertices.  We align horizontally and put
            # the lower number on the left vertex.
            G.set_pos(pos={0:[-0.5,0],1:[0.5,0]})
        else:
            maxdegree = max(max(N))
            if n == 3:
                # o--o--o
                centervert = [i for i in range(3) if max(N.row(i)) < maxdegree][0]
                other = [i for i in range(3) if i != centervert]
                G.set_pos(pos={centervert:[0,0],other[0]:[-1,0],other[1]:[1,0]})
            elif maxdegree == 4:
                # o--o<8
                centervert = [i for i in range(4) if max(N.row(i)) < maxdegree][0]
                other = [i for i in range(4) if i != centervert]
                G.set_pos(pos={centervert:[0,0],other[0]:[0,1],other[1]:[-0.8660254,-0.5],other[2]:[0.8660254,-0.5]})
            elif maxdegree == 27:
                # o--o--o--o
                centers = [i for i in range(4) if list(N.row(i)).count(3) == 2]
                left = [j for j in range(4) if N[centers[0],j] == 3 and j not in centers][0]
                right = [j for j in range(4) if N[centers[1],j] == 3 and j not in centers][0]
                G.set_pos(pos={left:[-1.5,0],centers[0]:[-0.5,0],centers[1]:[0.5,0],right:[1.5,0]})
            elif n == 4:
                # square
                opp = [i for i in range(1,4) if not N[0,i].is_prime()][0]
                other = [i for i in range(1,4) if i != opp]
                G.set_pos(pos={0:[1,1],other[0]:[-1,1],opp:[-1,-1],other[1]:[1,-1]})
            elif maxdegree == 8:
                # 8>o--o<8
                centers = [i for i in range(6) if list(N.row(i)).count(2) == 3]
                left = [j for j in range(6) if N[centers[0],j] == 2 and j not in centers]
                right = [j for j in range(6) if N[centers[1],j] == 2 and j not in centers]
                G.set_pos(pos={centers[0]:[-0.5,0],left[0]:[-1,0.8660254],left[1]:[-1,-0.8660254],centers[1]:[0.5,0],right[0]:[1,0.8660254],right[1]:[1,-0.8660254]})
            elif maxdegree == 18:
                # two squares joined on an edge
                centers = [i for i in range(6) if list(N.row(i)).count(3) == 2]
                top = [j for j in range(6) if N[centers[0],j] == 3]
                bl = [j for j in range(6) if N[top[0],j] == 2][0]
                br = [j for j in range(6) if N[top[1],j] == 2][0]
                G.set_pos(pos={centers[0]:[0,0.5],centers[1]:[0,-0.5],top[0]:[-1,0.5],top[1]:[1,0.5],bl:[-1,-0.5],br:[1,-0.5]})
            elif maxdegree == 16:
                # tree from bottom, 3 regular except for the leaves.
                centers = [i for i in range(8) if list(N.row(i)).count(2) == 3]
                center = [i for i in centers if len([j for j in centers if N[i,j] == 2]) == 2][0]
                centers.remove(center)
                bottom = [j for j in range(8) if N[center,j] == 2 and j not in centers][0]
                left = [j for j in range(8) if N[centers[0],j] == 2 and j != center]
                right = [j for j in range(8) if N[centers[1],j] == 2 and j != center]
                G.set_pos(pos={center:[0,0],bottom:[0,-1],centers[0]:[-0.8660254,0.5],centers[1]:[0.8660254,0.5],left[0]:[-0.8660254,1.5],right[0]:[0.8660254,1.5],left[1]:[-1.7320508,0],right[1]:[1.7320508,0]})
            elif maxdegree == 12:
                # tent
                centers = [i for i in range(8) if list(N.row(i)).count(2) == 3]
                left = [j for j in range(8) if N[centers[0],j] == 2]
                right = []
                for i in range(3):
                    right.append([j for j in range(8) if N[centers[1],j] == 2 and N[left[i],j] == 3][0])
                G.set_pos(pos={centers[0]:[-0.75,0],centers[1]:[0.75,0],left[0]:[-0.75,1],right[0]:[0.75,1],left[1]:[-1.25,-0.75],right[1]:[0.25,-0.75],left[2]:[-0.25,-0.25],right[2]:[1.25,-0.25]})
        G.set_vertices(D)
        G.relabel(range(1,n+1))
        return G

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import vertices [as 别名]
def _check_pbd(B,v,S):
    r"""
    Checks that ``B`` is a PBD on ``v`` points with given block sizes ``S``.

    The points of the balanced incomplete block design are implicitely assumed
    to be `\{0, ..., v-1\}`.

    INPUT:

    - ``B`` -- a list of blocks

    - ``v`` (integer) -- number of points

    - ``S`` -- list of integers `\geq 2`.

    EXAMPLE::

        sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
        [[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
         [0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
         [0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
        ...
        sage: from sage.combinat.designs.bibd import _check_pbd
        sage: _check_pbd([[1],[]],1,[1,0])
        Traceback (most recent call last):
        ...
        RuntimeError: All integers of S must be >=2

    TESTS::

        sage: _check_pbd([[1,2]],2,[2])
        Traceback (most recent call last):
        ...
        RuntimeError: The PBD covers a point 2 which is not in {0, 1}
        sage: _check_pbd([[1,2]]*2,2,[2])
        Traceback (most recent call last):
        ...
        RuntimeError: The pair (1,2) is covered more than once
        sage: _check_pbd([],2,[2])
        Traceback (most recent call last):
        ...
        RuntimeError: The pair (0,1) is not covered
        sage: _check_pbd([[1,2],[1]],2,[2])
        Traceback (most recent call last):
        ...
        RuntimeError: A block has size 1 while S=[2]
    """
    from itertools import combinations
    from sage.graphs.graph import Graph

    for X in B:
        if len(X) not in S:
            raise RuntimeError("A block has size {} while S={}".format(len(X),S))

    if any(x < 2 for x in S):
        raise RuntimeError("All integers of S must be >=2")

    if v == 0 or v == 1:
        if B:
            raise RuntimeError("A PBD with v<=1 is expected to be empty.")

    g = Graph()
    g.add_vertices(range(v))
    m = 0
    for X in B:
        for i,j in combinations(X,2):
            g.add_edge(i,j)
            m_tmp = g.size()
            if m_tmp != m+1:
                raise RuntimeError("The pair ({},{}) is covered more than once".format(i,j))
            m = m_tmp

    if g.vertices() != range(v):
        from sage.sets.integer_range import IntegerRange
        p = (set(g.vertices())-set(range(v))).pop()
        raise RuntimeError("The PBD covers a point {} which is not in {}".format(p,IntegerRange(v)))

    if not g.is_clique():
        for p1 in g:
            if g.degree(p1) != v-1:
                break
        neighbors = g.neighbors(p1)+[p1]
        p2 = (set(g.vertices())-set(neighbors)).pop()
        raise RuntimeError("The pair ({},{}) is not covered".format(p1,p2))

    return B