CoCalc -- sage_graphs_and_independence

Path: sage_graphs_and_independence_basics.sagews

Views: ¹⁸⁴

#GRAPHS IS A sAGE CLASS FOR CONSTRustors of several well-known graphs and graph classes
pete = graphs.PetersenGraph()

#.independent_set is a method for finding a maximum independent set, available for any graph object
pete.independent_set()

[0, 3, 6, 7]

len([1,2,3,4])

4

#define any indepence number function
#which inputs a graph g and outputs the independence number of g
def independence_number(g):
    return len(g.independent_set())

independence_number(pete)

4

show(pete)

d3-based renderer not yet implemented

#TO SEE THE CODE FOR A METHOD, USE ?? AFTER THE METHOD NAME
pete.independent_set??

   File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/graphs/graph.py
   Source:
       @doc_index("Algorithmically hard stuff")
    def independent_set(self, algorithm = "Cliquer", value_only = False, reduction_rules = True, solver = None, verbosity = 0):
        r"""
        Returns a maximum independent set.

        An independent set of a graph is a set of pairwise non-adjacent
        vertices. A maximum independent set is an independent set of maximum
        cardinality.  It induces an empty subgraph.

        Equivalently, an independent set is defined as the complement of a
        vertex cover.

        For more information, see the
        :wikipedia:`Independent_set_(graph_theory)` and the
        :wikipedia:`Vertex_cover`.

        INPUT:

        - ``algorithm`` -- the algorithm to be used

          * If ``algorithm = "Cliquer"`` (default), the problem is solved
            using Cliquer [NisOst2003]_.

            (see the :mod:`Cliquer modules <sage.graphs.cliquer>`)

          * If ``algorithm = "MILP"``, the problem is solved through a Mixed
            Integer Linear Program.

            (see :class:`~sage.numerical.mip.MixedIntegerLinearProgram`)

         * If ``algorithm = "mcqd"`` - Uses the MCQD solver
           (`<http://www.sicmm.org/~konc/maxclique/>`_). Note that the MCQD
           package must be installed.

        - ``value_only`` -- boolean (default: ``False``). If set to ``True``,
          only the size of a maximum independent set is returned. Otherwise,
          a maximum independent set is returned as a list of vertices.

        - ``reduction_rules`` -- (default: ``True``) Specify if the reductions
          rules from kernelization must be applied as pre-processing or not.
          See [ACFLSS04]_ for more details. Note that depending on the
          instance, it might be faster to disable reduction rules.

        - ``solver`` -- (default: ``None``) Specify a Linear Program (LP)
          solver to be used. If set to ``None``, the default one is used. For
          more information on LP solvers and which default solver is used, see
          the method
          :meth:`~sage.numerical.mip.MixedIntegerLinearProgram.solve`
          of the class
          :class:`~sage.numerical.mip.MixedIntegerLinearProgram`.

        - ``verbosity`` -- non-negative integer (default: ``0``). Set the level
          of verbosity you want from the linear program solver. Since the
          problem of computing an independent set is `NP`-complete, its solving
          may take some time depending on the graph. A value of 0 means that
          there will be no message printed by the solver. This option is only
          useful if ``algorithm="MILP"``.

        .. NOTE::

            While Cliquer/MCAD are usually (and by far) the most efficient
            implementations, the MILP formulation sometimes proves faster on
            very "symmetrical" graphs.

        EXAMPLES:

        Using Cliquer::

            sage: C = graphs.PetersenGraph()
            sage: C.independent_set()
            [0, 3, 6, 7]

        As a linear program::

            sage: C = graphs.PetersenGraph()
            sage: len(C.independent_set(algorithm = "MILP"))
            4

        .. PLOT::

            g = graphs.PetersenGraph()
            sphinx_plot(g.plot(partition=[g.independent_set()]))
        """
        my_cover = self.vertex_cover(algorithm=algorithm, value_only=value_only, reduction_rules=reduction_rules, solver=solver, verbosity=verbosity)
        if value_only:
            return self.order() - my_cover
        else:
            return [u for u in self.vertices() if not u in my_cover]

#complete graph on 20 vertices
k20 = graphs.CompleteGraph(20)
show(k20)

d3-based renderer not yet implemented

M = matrix(2,3,[1,2,3,4,5,6])
M

[1 2 3]
[4 5 6]

latex(M)

\left(\begin{array}{rrr}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right)

latex(k20)

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\Vertex[style={minimum size=1.0cm,draw=cv8,fill=cfv8,text=clv8,shape=circle},LabelOut=false,L=\hbox{$8$},x=1.0305cm,y=0.4775cm]{v8}
\Vertex[style={minimum size=1.0cm,draw=cv9,fill=cfv9,text=clv9,shape=circle},LabelOut=false,L=\hbox{$9$},x=1.7275cm,y=0.1224cm]{v9}
\Vertex[style={minimum size=1.0cm,draw=cv10,fill=cfv10,text=clv10,shape=circle},LabelOut=false,L=\hbox{$10$},x=2.5cm,y=0.0cm]{v10}
\Vertex[style={minimum size=1.0cm,draw=cv11,fill=cfv11,text=clv11,shape=circle},LabelOut=false,L=\hbox{$11$},x=3.2725cm,y=0.1224cm]{v11}
\Vertex[style={minimum size=1.0cm,draw=cv12,fill=cfv12,text=clv12,shape=circle},LabelOut=false,L=\hbox{$12$},x=3.9695cm,y=0.4775cm]{v12}
\Vertex[style={minimum size=1.0cm,draw=cv13,fill=cfv13,text=clv13,shape=circle},LabelOut=false,L=\hbox{$13$},x=4.5225cm,y=1.0305cm]{v13}
\Vertex[style={minimum size=1.0cm,draw=cv14,fill=cfv14,text=clv14,shape=circle},LabelOut=false,L=\hbox{$14$},x=4.8776cm,y=1.7275cm]{v14}
\Vertex[style={minimum size=1.0cm,draw=cv15,fill=cfv15,text=clv15,shape=circle},LabelOut=false,L=\hbox{$15$},x=5.0cm,y=2.5cm]{v15}
\Vertex[style={minimum size=1.0cm,draw=cv16,fill=cfv16,text=clv16,shape=circle},LabelOut=false,L=\hbox{$16$},x=4.8776cm,y=3.2725cm]{v16}
\Vertex[style={minimum size=1.0cm,draw=cv17,fill=cfv17,text=clv17,shape=circle},LabelOut=false,L=\hbox{$17$},x=4.5225cm,y=3.9695cm]{v17}
\Vertex[style={minimum size=1.0cm,draw=cv18,fill=cfv18,text=clv18,shape=circle},LabelOut=false,L=\hbox{$18$},x=3.9695cm,y=4.5225cm]{v18}
\Vertex[style={minimum size=1.0cm,draw=cv19,fill=cfv19,text=clv19,shape=circle},LabelOut=false,L=\hbox{$19$},x=3.2725cm,y=4.8776cm]{v19}
%
\Edge[lw=0.1cm,style={color=cv0v1,},](v0)(v1)
\Edge[lw=0.1cm,style={color=cv0v2,},](v0)(v2)
\Edge[lw=0.1cm,style={color=cv0v3,},](v0)(v3)
\Edge[lw=0.1cm,style={color=cv0v4,},](v0)(v4)
\Edge[lw=0.1cm,style={color=cv0v5,},](v0)(v5)
\Edge[lw=0.1cm,style={color=cv0v6,},](v0)(v6)
\Edge[lw=0.1cm,style={color=cv0v7,},](v0)(v7)
\Edge[lw=0.1cm,style={color=cv0v8,},](v0)(v8)
\Edge[lw=0.1cm,style={color=cv0v9,},](v0)(v9)
\Edge[lw=0.1cm,style={color=cv0v10,},](v0)(v10)
\Edge[lw=0.1cm,style={color=cv0v11,},](v0)(v11)
\Edge[lw=0.1cm,style={color=cv0v12,},](v0)(v12)
\Edge[lw=0.1cm,style={color=cv0v13,},](v0)(v13)
\Edge[lw=0.1cm,style={color=cv0v14,},](v0)(v14)
\Edge[lw=0.1cm,style={color=cv0v15,},](v0)(v15)
\Edge[lw=0.1cm,style={color=cv0v16,},](v0)(v16)
\Edge[lw=0.1cm,style={color=cv0v17,},](v0)(v17)
\Edge[lw=0.1cm,style={color=cv0v18,},](v0)(v18)
\Edge[lw=0.1cm,style={color=cv0v19,},](v0)(v19)
\Edge[lw=0.1cm,style={color=cv1v2,},](v1)(v2)
\Edge[lw=0.1cm,style={color=cv1v3,},](v1)(v3)
\Edge[lw=0.1cm,style={color=cv1v4,},](v1)(v4)
\Edge[lw=0.1cm,style={color=cv1v5,},](v1)(v5)
\Edge[lw=0.1cm,style={color=cv1v6,},](v1)(v6)
\Edge[lw=0.1cm,style={color=cv1v7,},](v1)(v7)
\Edge[lw=0.1cm,style={color=cv1v8,},](v1)(v8)
\Edge[lw=0.1cm,style={color=cv1v9,},](v1)(v9)
\Edge[lw=0.1cm,style={color=cv1v10,},](v1)(v10)
\Edge[lw=0.1cm,style={color=cv1v11,},](v1)(v11)
\Edge[lw=0.1cm,style={color=cv1v12,},](v1)(v12)
\Edge[lw=0.1cm,style={color=cv1v13,},](v1)(v13)
\Edge[lw=0.1cm,style={color=cv1v14,},](v1)(v14)
\Edge[lw=0.1cm,style={color=cv1v15,},](v1)(v15)
\Edge[lw=0.1cm,style={color=cv1v16,},](v1)(v16)
\Edge[lw=0.1cm,style={color=cv1v17,},](v1)(v17)
\Edge[lw=0.1cm,style={color=cv1v18,},](v1)(v18)
\Edge[lw=0.1cm,style={color=cv1v19,},](v1)(v19)
\Edge[lw=0.1cm,style={color=cv2v3,},](v2)(v3)
\Edge[lw=0.1cm,style={color=cv2v4,},](v2)(v4)
\Edge[lw=0.1cm,style={color=cv2v5,},](v2)(v5)
\Edge[lw=0.1cm,style={color=cv2v6,},](v2)(v6)
\Edge[lw=0.1cm,style={color=cv2v7,},](v2)(v7)
\Edge[lw=0.1cm,style={color=cv2v8,},](v2)(v8)
\Edge[lw=0.1cm,style={color=cv2v9,},](v2)(v9)
\Edge[lw=0.1cm,style={color=cv2v10,},](v2)(v10)
\Edge[lw=0.1cm,style={color=cv2v11,},](v2)(v11)
\Edge[lw=0.1cm,style={color=cv2v12,},](v2)(v12)
\Edge[lw=0.1cm,style={color=cv2v13,},](v2)(v13)
\Edge[lw=0.1cm,style={color=cv2v14,},](v2)(v14)
\Edge[lw=0.1cm,style={color=cv2v15,},](v2)(v15)
\Edge[lw=0.1cm,style={color=cv2v16,},](v2)(v16)
\Edge[lw=0.1cm,style={color=cv2v17,},](v2)(v17)
\Edge[lw=0.1cm,style={color=cv2v18,},](v2)(v18)
\Edge[lw=0.1cm,style={color=cv2v19,},](v2)(v19)
\Edge[lw=0.1cm,style={color=cv3v4,},](v3)(v4)
\Edge[lw=0.1cm,style={color=cv3v5,},](v3)(v5)
\Edge[lw=0.1cm,style={color=cv3v6,},](v3)(v6)
\Edge[lw=0.1cm,style={color=cv3v7,},](v3)(v7)
\Edge[lw=0.1cm,style={color=cv3v8,},](v3)(v8)
\Edge[lw=0.1cm,style={color=cv3v9,},](v3)(v9)
\Edge[lw=0.1cm,style={color=cv3v10,},](v3)(v10)
\Edge[lw=0.1cm,style={color=cv3v11,},](v3)(v11)
\Edge[lw=0.1cm,style={color=cv3v12,},](v3)(v12)
\Edge[lw=0.1cm,style={color=cv3v13,},](v3)(v13)
\Edge[lw=0.1cm,style={color=cv3v14,},](v3)(v14)
\Edge[lw=0.1cm,style={color=cv3v15,},](v3)(v15)
\Edge[lw=0.1cm,style={color=cv3v16,},](v3)(v16)
\Edge[lw=0.1cm,style={color=cv3v17,},](v3)(v17)
\Edge[lw=0.1cm,style={color=cv3v18,},](v3)(v18)
\Edge[lw=0.1cm,style={color=cv3v19,},](v3)(v19)
\Edge[lw=0.1cm,style={color=cv4v5,},](v4)(v5)
\Edge[lw=0.1cm,style={color=cv4v6,},](v4)(v6)
\Edge[lw=0.1cm,style={color=cv4v7,},](v4)(v7)
\Edge[lw=0.1cm,style={color=cv4v8,},](v4)(v8)
\Edge[lw=0.1cm,style={color=cv4v9,},](v4)(v9)
\Edge[lw=0.1cm,style={color=cv4v10,},](v4)(v10)
\Edge[lw=0.1cm,style={color=cv4v11,},](v4)(v11)
\Edge[lw=0.1cm,style={color=cv4v12,},](v4)(v12)
\Edge[lw=0.1cm,style={color=cv4v13,},](v4)(v13)
\Edge[lw=0.1cm,style={color=cv4v14,},](v4)(v14)
\Edge[lw=0.1cm,style={color=cv4v15,},](v4)(v15)
\Edge[lw=0.1cm,style={color=cv4v16,},](v4)(v16)
\Edge[lw=0.1cm,style={color=cv4v17,},](v4)(v17)
\Edge[lw=0.1cm,style={color=cv4v18,},](v4)(v18)
\Edge[lw=0.1cm,style={color=cv4v19,},](v4)(v19)
\Edge[lw=0.1cm,style={color=cv5v6,},](v5)(v6)
\Edge[lw=0.1cm,style={color=cv5v7,},](v5)(v7)
\Edge[lw=0.1cm,style={color=cv5v8,},](v5)(v8)
\Edge[lw=0.1cm,style={color=cv5v9,},](v5)(v9)
\Edge[lw=0.1cm,style={color=cv5v10,},](v5)(v10)
\Edge[lw=0.1cm,style={color=cv5v11,},](v5)(v11)
\Edge[lw=0.1cm,style={color=cv5v12,},](v5)(v12)
\Edge[lw=0.1cm,style={color=cv5v13,},](v5)(v13)
\Edge[lw=0.1cm,style={color=cv5v14,},](v5)(v14)
\Edge[lw=0.1cm,style={color=cv5v15,},](v5)(v15)
\Edge[lw=0.1cm,style={color=cv5v16,},](v5)(v16)
\Edge[lw=0.1cm,style={color=cv5v17,},](v5)(v17)
\Edge[lw=0.1cm,style={color=cv5v18,},](v5)(v18)
\Edge[lw=0.1cm,style={color=cv5v19,},](v5)(v19)
\Edge[lw=0.1cm,style={color=cv6v7,},](v6)(v7)
\Edge[lw=0.1cm,style={color=cv6v8,},](v6)(v8)
\Edge[lw=0.1cm,style={color=cv6v9,},](v6)(v9)
\Edge[lw=0.1cm,style={color=cv6v10,},](v6)(v10)
\Edge[lw=0.1cm,style={color=cv6v11,},](v6)(v11)
\Edge[lw=0.1cm,style={color=cv6v12,},](v6)(v12)
\Edge[lw=0.1cm,style={color=cv6v13,},](v6)(v13)
\Edge[lw=0.1cm,style={color=cv6v14,},](v6)(v14)
\Edge[lw=0.1cm,style={color=cv6v15,},](v6)(v15)
\Edge[lw=0.1cm,style={color=cv6v16,},](v6)(v16)
\Edge[lw=0.1cm,style={color=cv6v17,},](v6)(v17)
\Edge[lw=0.1cm,style={color=cv6v18,},](v6)(v18)
\Edge[lw=0.1cm,style={color=cv6v19,},](v6)(v19)
\Edge[lw=0.1cm,style={color=cv7v8,},](v7)(v8)
\Edge[lw=0.1cm,style={color=cv7v9,},](v7)(v9)
\Edge[lw=0.1cm,style={color=cv7v10,},](v7)(v10)
\Edge[lw=0.1cm,style={color=cv7v11,},](v7)(v11)
\Edge[lw=0.1cm,style={color=cv7v12,},](v7)(v12)
\Edge[lw=0.1cm,style={color=cv7v13,},](v7)(v13)
\Edge[lw=0.1cm,style={color=cv7v14,},](v7)(v14)
\Edge[lw=0.1cm,style={color=cv7v15,},](v7)(v15)
\Edge[lw=0.1cm,style={color=cv7v16,},](v7)(v16)
\Edge[lw=0.1cm,style={color=cv7v17,},](v7)(v17)
\Edge[lw=0.1cm,style={color=cv7v18,},](v7)(v18)
\Edge[lw=0.1cm,style={color=cv7v19,},](v7)(v19)
\Edge[lw=0.1cm,style={color=cv8v9,},](v8)(v9)
\Edge[lw=0.1cm,style={color=cv8v10,},](v8)(v10)
\Edge[lw=0.1cm,style={color=cv8v11,},](v8)(v11)
\Edge[lw=0.1cm,style={color=cv8v12,},](v8)(v12)
\Edge[lw=0.1cm,style={color=cv8v13,},](v8)(v13)
\Edge[lw=0.1cm,style={color=cv8v14,},](v8)(v14)
\Edge[lw=0.1cm,style={color=cv8v15,},](v8)(v15)
\Edge[lw=0.1cm,style={color=cv8v16,},](v8)(v16)
\Edge[lw=0.1cm,style={color=cv8v17,},](v8)(v17)
\Edge[lw=0.1cm,style={color=cv8v18,},](v8)(v18)
\Edge[lw=0.1cm,style={color=cv8v19,},](v8)(v19)
\Edge[lw=0.1cm,style={color=cv9v10,},](v9)(v10)
\Edge[lw=0.1cm,style={color=cv9v11,},](v9)(v11)
\Edge[lw=0.1cm,style={color=cv9v12,},](v9)(v12)
\Edge[lw=0.1cm,style={color=cv9v13,},](v9)(v13)
\Edge[lw=0.1cm,style={color=cv9v14,},](v9)(v14)
\Edge[lw=0.1cm,style={color=cv9v15,},](v9)(v15)
\Edge[lw=0.1cm,style={color=cv9v16,},](v9)(v16)
\Edge[lw=0.1cm,style={color=cv9v17,},](v9)(v17)
\Edge[lw=0.1cm,style={color=cv9v18,},](v9)(v18)
\Edge[lw=0.1cm,style={color=cv9v19,},](v9)(v19)
\Edge[lw=0.1cm,style={color=cv10v11,},](v10)(v11)
\Edge[lw=0.1cm,style={color=cv10v12,},](v10)(v12)
\Edge[lw=0.1cm,style={color=cv10v13,},](v10)(v13)
\Edge[lw=0.1cm,style={color=cv10v14,},](v10)(v14)
\Edge[lw=0.1cm,style={color=cv10v15,},](v10)(v15)
\Edge[lw=0.1cm,style={color=cv10v16,},](v10)(v16)
\Edge[lw=0.1cm,style={color=cv10v17,},](v10)(v17)
\Edge[lw=0.1cm,style={color=cv10v18,},](v10)(v18)
\Edge[lw=0.1cm,style={color=cv10v19,},](v10)(v19)
\Edge[lw=0.1cm,style={color=cv11v12,},](v11)(v12)
\Edge[lw=0.1cm,style={color=cv11v13,},](v11)(v13)
\Edge[lw=0.1cm,style={color=cv11v14,},](v11)(v14)
\Edge[lw=0.1cm,style={color=cv11v15,},](v11)(v15)
\Edge[lw=0.1cm,style={color=cv11v16,},](v11)(v16)
\Edge[lw=0.1cm,style={color=cv11v17,},](v11)(v17)
\Edge[lw=0.1cm,style={color=cv11v18,},](v11)(v18)
\Edge[lw=0.1cm,style={color=cv11v19,},](v11)(v19)
\Edge[lw=0.1cm,style={color=cv12v13,},](v12)(v13)
\Edge[lw=0.1cm,style={color=cv12v14,},](v12)(v14)
\Edge[lw=0.1cm,style={color=cv12v15,},](v12)(v15)
\Edge[lw=0.1cm,style={color=cv12v16,},](v12)(v16)
\Edge[lw=0.1cm,style={color=cv12v17,},](v12)(v17)
\Edge[lw=0.1cm,style={color=cv12v18,},](v12)(v18)
\Edge[lw=0.1cm,style={color=cv12v19,},](v12)(v19)
\Edge[lw=0.1cm,style={color=cv13v14,},](v13)(v14)
\Edge[lw=0.1cm,style={color=cv13v15,},](v13)(v15)
\Edge[lw=0.1cm,style={color=cv13v16,},](v13)(v16)
\Edge[lw=0.1cm,style={color=cv13v17,},](v13)(v17)
\Edge[lw=0.1cm,style={color=cv13v18,},](v13)(v18)
\Edge[lw=0.1cm,style={color=cv13v19,},](v13)(v19)
\Edge[lw=0.1cm,style={color=cv14v15,},](v14)(v15)
\Edge[lw=0.1cm,style={color=cv14v16,},](v14)(v16)
\Edge[lw=0.1cm,style={color=cv14v17,},](v14)(v17)
\Edge[lw=0.1cm,style={color=cv14v18,},](v14)(v18)
\Edge[lw=0.1cm,style={color=cv14v19,},](v14)(v19)
\Edge[lw=0.1cm,style={color=cv15v16,},](v15)(v16)
\Edge[lw=0.1cm,style={color=cv15v17,},](v15)(v17)
\Edge[lw=0.1cm,style={color=cv15v18,},](v15)(v18)
\Edge[lw=0.1cm,style={color=cv15v19,},](v15)(v19)
\Edge[lw=0.1cm,style={color=cv16v17,},](v16)(v17)
\Edge[lw=0.1cm,style={color=cv16v18,},](v16)(v18)
\Edge[lw=0.1cm,style={color=cv16v19,},](v16)(v19)
\Edge[lw=0.1cm,style={color=cv17v18,},](v17)(v18)
\Edge[lw=0.1cm,style={color=cv17v19,},](v17)(v19)
\Edge[lw=0.1cm,style={color=cv18v19,},](v18)(v19)
%
\end{tikzpicture}

pete = graphs.PetersenGraph()

pete.degree()

[3, 3, 3, 3, 3, 3, 3, 3, 3, 3]

def min_degree(g):
    return min(g.degree())
def max_degree(g):
    return max(g.degree())

min_degree(pete)

3

max_degree(pete)

3

#an example of coding in a graph by hand
#triangle with a single pendant vertex

triangle_with_pendant = Graph(4)
triangle_with_pendant.add_edge(0,1)
triangle_with_pendant.add_edge(2,1)
triangle_with_pendant.add_edge(2,3)
triangle_with_pendant.add_edge(1,3)

show(triangle_with_pendant)

d3-based renderer not yet implemented

independence_number(triangle_with_pendant)

2

test_graph = Graph(5)

#testing if two graphs are the same, that is, they are "isomorphic"
k5 = graphs.CompleteGraph(5)

test = graphs.CompleteGraph(5)

k5.vertices()

[0, 1, 2, 3, 4]

k5.is_isomorphic(test)

True

#here's how to get a useful "name" of a graph (called a g6 string)
k5.graph6_string()

'D~{'

#reconstructing a graph from its g6 string
recon_k5 = Graph('D~{')

show(recon_k5)

d3-based renderer not yet implemented

load("gt.sage")
for g in graph_objects[50:100]:
    print g.name()
    print g.graph6_string()
    print "order is {}, size is {}, independence number is {}".format(g.order(),g.size(), independence_number(g))
    g.show()

loaded graph dc1024
loaded graph dc2048
loaded graph properties data file
loaded graph invariants data file
Octahedron
E}lw
order is 6, size is 12, independence number is 2

Thomsen graph
EFz_
order is 6, size is 9, independence number is 3

Petersen graph
IheA@GUAo
order is 10, size is 15, independence number is 4

Pappus Graph
QhEKA?_C?O?_?P?g?I?@S?DOAG_
order is 18, size is 27, independence number is 9

Grotzsch graph
Jsa@IchDIS_
order is 11, size is 20, independence number is 5

Gray graph
uhCGGD@?G?_@?@??_GG?@C?C??G??G??C?@@???G?_?_??@???@????_??GG???@??C?E????GG???G????C???@@?????G???_?_O???@?@???@??????_????GG?????@????C?C?A????G??G???G??????C?????@@A??????G?????_?_??O???@???@???@G???????_??????GG?O?????@??????C?E???A????G
order is 54, size is 81, independence number is 27

Heawood graph
MhEGHC@AI?_PC@_G_
order is 14, size is 21, independence number is 7

Herschel graph
Jl`I@CPE?L_
order is 11, size is 18, independence number is 6

Coxeter Graph
[hCGGC@?G?_`O@??_?G?@??CG?G??G?AC?O@C??G???o??@_O?_GGC??@@?_C@?O
order is 28, size is 42, independence number is 12

Brinkmann graph
TQIQU@_K?o@_@___?CGA_?I??S_?T_?Io@AK
order is 21, size is 42, independence number is 7

Tutte-Coxeter graph
]hCGGC@GG?_@?@A?_?G@@??E??GG?G?OC??@??GI???_O?@?@?@??A?a???G??@@?O??E?A??G
order is 30, size is 45, independence number is 15

Tutte Graph
mHCKGC@OG??D?DG?`@GIOC????G??G??C??@_??G???_??@??A@????????g???D??@?C??@@G??@Q???C??????@O????G?????_????@_????@??????_?????G????A@?????????????g??????g????@?C?????GH?????@Q?
order is 46, size is 69, independence number is 19

Robertson Graph
RhDGGe@GGG_Ha@G__@GP@O@CAOK_@G
order is 19, size is 38, independence number is 7

Folkman Graph
ShEGGCPIG__@?P?ggGL?@O@C?IGGGKS?C
order is 20, size is 40, independence number is 10

Balaban 10-cage
~?@EhCGGC@?G?_@?@??_?G?@??CG?G??G?OC??@???G???_??@???@??A?_???G@??@????C????G?A??G???OCO???@?????G?_???_????@??C??@??????_??A??G?????@@?????C?A????G??????G???@??C??????@???G???I???????_?C????@?O?????@????????_???????G???C???@?@??????C_???????G?????A??G????????C????C???@???_?????G???????A?_??????O?@?????@???@_?????????_???_?????GC????????@?????????OC???????_??G?????_????G???????@??C??????O???@_???A??????G
order is 70, size is 105, independence number is 35

Pappus Graph
QhEKA?_C?O?_?P?g?I?@S?DOAG_
order is 18, size is 27, independence number is 9

Tietze Graph
KhDGHEH_?__R
order is 12, size is 18, independence number is 5

Sylvester Graph
c_Go??A?o????A?B???????G??oaG_PCU?QOG?QGc?GhH?@`OA??I@?cA_?I_D?@a@@??EA??GAB??J@?_?EO`???ACG??GGO_??gO`??@
order is 36, size is 90, independence number is 12

Szekeres Snark Graph
q??CGC@_GG_@_@?G_?A?@??C??I??G?@C??@O??GG?G_???G??@?@??_???H???@???@C????H????G?@?@C?????C????G??@??_????@C????@?????G_G????G_????@???@?@C?@?????O?????G????G?C??????@A??????G??????G_?@????@A??????@?????G?G_
order is 50, size is 75, independence number is 21

Moebius-Kantor Graph
OhCGKE?O@?ACAC@I?Q_AS
order is 16, size is 24, independence number is 8

ryan
WxEW?CB?I?_R????_?W?@?OC?AW???O?C??B???G?A?_??R
order is 24, size is 36, independence number is 8

inp
J?`FBo{fdb?
order is 11, size is 23, independence number is 4

c4c4
FlCGg
order is 7, size is 8, independence number is 4

regular_non_trans
G^Q?W[
order is 8, size is 12, independence number is 3

bridge
DU{
order is 5, size is 7, independence number is 2

p10k4
MhCGGC@?G?_@_B?B_
order is 14, size is 17, independence number is 6

c100
~?@csP@@?OC?O`?@?@_?O?A??W??_??_G?O??C??@_??C???G???G@??K???A????O???@????A????A?G??B?????_????C?G???O????@_?????_?????O?????C?G???@_?????E??????G??????G?G????C??????@???????G???????o??????@???????@????????_?_?????W???????@????????C????????G????????G?G??????E????????@_????????K?????????_????????@?@???????@?@???????@_?????????G?????????@?@????????C?C????????W??????????W??????????C??????????@?@?????????G???????????_??????????@?@??????????_???????????O???????????C?G??????????O???????????@????????????A????????????A?G??????????@_????????????W????????????@_????????????E?????????????E?????????????E?????????????B??????????????O?????????????A@?????????????G??????????????OG?????????????O??????????????GC?????????????A???????????????OG?????????????@?_?????????????B???????????????@_???????????????W???????????????@_???????????????F
order is 100, size is 150, independence number is 43

starfish
N~~eeQoiCoM?Y?U?F??
order is 15, size is 40, independence number is 10

c5k3
FheCG
order is 7, size is 8, independence number is 3

k5pendant
E~}?
order is 6, size is 11, independence number is 2

Shrikhande graph
OzK[]L@gA`DEDEAj?msCu
order is 16, size is 48, independence number is 4

sylvester
Olw?GCD@o??@?@?A_@o`A
order is 16, size is 24, independence number is 7

fork
DhO
order is 5, size is 4, independence number is 3

edge_critical_5
Dls
order is 5, size is 7, independence number is 2

edge_critical_11_1
JxCIGCP@K?_
order is 11, size is 15, independence number is 5

edge_critical_11_2
JxCGGc@AKC_
order is 11, size is 15, independence number is 5

pete_minus
HheA@GU
order is 9, size is 12, independence number is 4

c5
Dhc
order is 5, size is 5, independence number is 2

bow_tie
D{c
order is 5, size is 6, independence number is 2

pepper_residue_graph
LJqc_?@?[AOC_C
order is 13, size is 18, independence number is 6

barrus_graph
HxNEG{W
order is 9, size is 17, independence number is 3

p5
DhC
order is 5, size is 4, independence number is 3

c6
EhEG
order is 6, size is 6, independence number is 3

c9
HhCGGE@
order is 9, size is 9, independence number is 4

ce3
Gr`HOk
order is 8, size is 12, independence number is 4

ce4
G~sNp?
order is 8, size is 16, independence number is 3

ce5
X~}AHKVB{GGPGRCJ`B{GOO`C`AW`AwO`}CGOO`AHACHaCGVACG^
order is 25, size is 100, independence number is 5

k4e2split
E~oo
order is 6, size is 10, independence number is 2

flower_with_3_petals
F{eCG
order is 7, size is 9, independence number is 3

flower_with_4_petals
H{eCKA@
order is 9, size is 12, independence number is 4

#IS a graph g already in gt.sage?
#NOTE the coded graphs are all in the list graph_objects
load("gt.sage")

loaded graph dc1024
loaded graph dc2048
loaded graph properties data file
loaded graph invariants data file

#look at all graphs with 9 vertices and 12 edges
#idea: can do this simple visual check *first* before coding anything
for g in graph_objects:
    if g.order() == 9 and g.size() == 12:
        g.show()

        g.show()

s3

s3: Graph on 4 vertices

s3.show()


nb = Graph('x??C?O?????A?@_G?H??????A?C??EGo?@S?O@?O??@G???CO???CAC_??a?@G?????H???????????O?_?H??G??G??@??_??OA?OCHCO?YA????????A?O???G?O?@????OOC???_@??????MCOC???O_??[Q??@???????O??_G?P?GO@A?G_???A???A@??g???W???@CG_???`_@O??????@?O@?AGO?????C??A??F??????@C????A?E@L?????P@`??')

nb.order()

57

nb.show()

order_automorphism_group(nb)

1

independence_number(nb)

25

nb = Graph("N~~eeQoiCoM?Y?U?F??")

nb.lovasz_theta()

9.9999999