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# Graph database

INFO:

This module implements classes (GraphDatabase, GraphQuery, GenericGraphQuery) for interfacing with the sqlite database graphs.db.

The GraphDatabase class interfaces with the sqlite database graphs.db. It is an immutable database that inherits from SQLDatabase (see sage.databases.database.py).

The database contains all unlabeled graphs with 7 or fewer nodes. This class will also interface with the optional database package containing all unlabeled graphs with 8 or fewer nodes. The database(s) consists of five tables, and has the structure given by the function graph_info. (For a full description including column data types, create a GraphDatabase instance and call the method get_skeleton).

AUTHORS:

- Emily A. Kirkman (2008-09-20): first version of interactive queries, cleaned up code and generalized many elements to sage.databases.database.py
- Emily A. Kirkman (2007-07-23): inherits GenericSQLDatabase, also added classes: GraphQuery and GenericGraphQuery
- Emily A. Kirkman (2007-05-11): initial sqlite version
- Emily A. Kirkman (2007-02-13): initial version (non-sqlite)

REFERENCES:

- Data provided by Jason Grout (Brigham Young University). [Online] Available: http://artsci.drake.edu/grout/graphs/

class `sage.graphs.graph_database.``GenericGraphQuery`(query_string, database=None, param_tuple=None)

  Bases: `sage.databases.sql_db.SQLQuery`

  A query for a GraphDatabase.

  INPUT:

  - `database` - the GraphDatabase instance to query (if None then a new instance is created)
  - `query_string` - a string representing the SQL query
  - `param_tuple` - a tuple of strings - what to replace question marks in query_string with (optional, but a good idea)

  Note

  This query class is generally intended for developers and more advanced users. It allows you to execute any query, and so may be considered unsafe.

  See GraphDatabase class docstrings or enter:

      sage: G = GraphDatabase()
      sage: G.get_skeleton()
      {...

  to see the underlying structure of the database. Also see SQLQuery in sage.databases.database for more info and a tutorial.

  A piece of advice about ‘?’ and param_tuple: It is generally considered safer to query with a ‘?’ in place of each value parameter, and using a second argument (a tuple of strings) in a call to the sqlite database. Successful use of the param_tuple argument is exemplified:

      sage: G = GraphDatabase()
      sage: q = 'select graph_id,graph6,num_vertices,num_edges from graph_data where graph_id<=(?) and num_vertices=(?)'
      sage: param = (22,5)
      sage: Q = SQLQuery(G,q,param)
      sage: Q.show()
      graph_id             graph6               num_vertices         num_edges
      --------------------------------------------------------------------------------
      18                   D??                  5                    0
      19                   D?C                  5                    1
      20                   D?K                  5                    2
      21                   D@O                  5                    2
      22                   D?[                  5                    3

class `sage.graphs.graph_database.``GraphDatabase`

  Bases: `sage.databases.sql_db.SQLDatabase`

  Graph Database

  INFO:

  This class interfaces with the sqlite database graphs.db. It is an immutable database that inherits from SQLDatabase (see sage.databases.database.py). The display functions and get_graphs_list create their own queries, but it is also possible to query the database by constructing either a SQLQuery.

  The database contains all unlabeled graphs with 7 or fewer nodes. This class will also interface with the optional database package containing all unlabeled graphs with 8 or fewer nodes. The database consists of five tables. For a full table and column structure, call graph_db_info.

  USE: The tables are associated by the unique primary key graph_id (int).

  To query this database, we create a GraphQuery. This can be done directly with the query method or by initializing one of 1. GenericGraphQuery - allows direct entry of a query string and tuple of parameters. This is the route for more advanced users that are familiar with SQL. 2. GraphQuery - is a wrapper of SQLQuery, a general database/query wrapper of SQLite for new users.

  REFERENCES:

  - Data provided by Jason Grout (Brigham Young University). [Online] Available: http://artsci.drake.edu/grout/graphs/

  EXAMPLE:

      sage: G = GraphDatabase()
      sage: G.get_skeleton()
      {u'aut_grp': {u'aut_grp_size': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'edge_transitive': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False},
        u'graph_id': {'index': False,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_fixed_points': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_orbits': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'vertex_transitive': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False}},
       u'degrees': {u'average_degree': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'degree_sequence': {'index': False,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'degrees_sd': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'graph_id': {'index': False,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'max_degree': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'min_degree': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'regular': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False}},
       u'graph_data': {u'complement_graph6': {'index': True,
         'primary_key': False,
         'sql': u'TEXT',
         'unique': False},
        u'eulerian': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False},
        u'graph6': {'index': True,
         'primary_key': False,
         'sql': u'TEXT',
         'unique': False},
        u'graph_id': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': True},
        u'lovasz_number': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'num_cycles': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_edges': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_hamiltonian_cycles': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_vertices': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'perfect': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False},
        u'planar': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False}},
       u'misc': {u'clique_number': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'diameter': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'edge_connectivity': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False},
        u'girth': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'graph_id': {'index': False,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'independence_number': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'induced_subgraphs': {'index': True,
         'primary_key': False,
         'sql': u'TEXT',
         'unique': False},
        u'min_vertex_cover_size': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_components': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_cut_vertices': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'num_spanning_trees': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'radius': {'index': True,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'vertex_connectivity': {'index': True,
         'primary_key': False,
         'sql': u'BOOLEAN',
         'unique': False}},
       u'spectrum': {u'eigenvalues_sd': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'energy': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'graph_id': {'index': False,
         'primary_key': False,
         'sql': u'INTEGER',
         'unique': False},
        u'max_eigenvalue': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'min_eigenvalue': {'index': True,
         'primary_key': False,
         'sql': u'REAL',
         'unique': False},
        u'spectrum': {'index': False,
         'primary_key': False,
         'sql': u'TEXT',
         'unique': False}}}

  `interactive_query`(display_cols, **kwds)

    TODO: This function could use improvement. Add full options of typical GraphQuery (i.e.: have it accept list input); and update options in interact to make it less annoying to put in operators.

    Generates an interact shell (in the notebook only) that allows the user to manipulate query parameters and see the updated results.

    EXAMPLE:

        sage: D = GraphDatabase()
        sage: D.interactive_query(display_cols=['graph6','num_vertices','degree_sequence'],num_edges=5,max_degree=3)
        <html>...</html>

  `query`(query_dict=None, display_cols=None, **kwds)

    Creates a GraphQuery on this database. For full class details, type GraphQuery? and press shift+enter.

    EXAMPLE:

        sage: D = GraphDatabase()
        sage: q = D.query(display_cols=['graph6','num_vertices','degree_sequence'],num_edges=['<=',5])
        sage: q.show()
        Graph6               Num Vertices         Degree Sequence
        ------------------------------------------------------------
        @                    1                    [0]
        A?                   2                    [0, 0]
        A_                   2                    [1, 1]
        B?                   3                    [0, 0, 0]
        BG                   3                    [0, 1, 1]
        BW                   3                    [1, 1, 2]
        Bw                   3                    [2, 2, 2]
        C?                   4                    [0, 0, 0, 0]
        C@                   4                    [0, 0, 1, 1]
        CB                   4                    [0, 1, 1, 2]
        CF                   4                    [1, 1, 1, 3]
        CJ                   4                    [0, 2, 2, 2]
        CK                   4                    [1, 1, 1, 1]
        CL                   4                    [1, 1, 2, 2]
        CN                   4                    [1, 2, 2, 3]
        C]                   4                    [2, 2, 2, 2]
        C^                   4                    [2, 2, 3, 3]
        D??                  5                    [0, 0, 0, 0, 0]
        D?C                  5                    [0, 0, 0, 1, 1]
        D?K                  5                    [0, 0, 1, 1, 2]
        D?[                  5                    [0, 1, 1, 1, 3]
        D?{                  5                    [1, 1, 1, 1, 4]
        D@K                  5                    [0, 0, 2, 2, 2]
        D@O                  5                    [0, 1, 1, 1, 1]
        D@S                  5                    [0, 1, 1, 2, 2]
        D@[                  5                    [0, 1, 2, 2, 3]
        D@s                  5                    [1, 1, 1, 2, 3]
        D@{                  5                    [1, 1, 2, 2, 4]
        DBW                  5                    [0, 2, 2, 2, 2]
        DB[                  5                    [0, 2, 2, 3, 3]
        DBg                  5                    [1, 1, 2, 2, 2]
        DBk                  5                    [1, 1, 2, 3, 3]
        DIk                  5                    [1, 2, 2, 2, 3]
        DK[                  5                    [1, 2, 2, 2, 3]
        DLo                  5                    [2, 2, 2, 2, 2]
        D_K                  5                    [1, 1, 1, 1, 2]
        D`K                  5                    [1, 1, 2, 2, 2]
        E???                 6                    [0, 0, 0, 0, 0, 0]
        E??G                 6                    [0, 0, 0, 0, 1, 1]
        E??W                 6                    [0, 0, 0, 1, 1, 2]
        E??w                 6                    [0, 0, 1, 1, 1, 3]
        E?@w                 6                    [0, 1, 1, 1, 1, 4]
        E?Bw                 6                    [1, 1, 1, 1, 1, 5]
        E?CW                 6                    [0, 0, 0, 2, 2, 2]
        E?C_                 6                    [0, 0, 1, 1, 1, 1]
        E?Cg                 6                    [0, 0, 1, 1, 2, 2]
        E?Cw                 6                    [0, 0, 1, 2, 2, 3]
        E?Dg                 6                    [0, 1, 1, 1, 2, 3]
        E?Dw                 6                    [0, 1, 1, 2, 2, 4]
        E?Fg                 6                    [1, 1, 1, 1, 2, 4]
        E?Ko                 6                    [0, 0, 2, 2, 2, 2]
        E?Kw                 6                    [0, 0, 2, 2, 3, 3]
        E?LO                 6                    [0, 1, 1, 2, 2, 2]
        E?LW                 6                    [0, 1, 1, 2, 3, 3]
        E?N?                 6                    [1, 1, 1, 1, 2, 2]
        E?NG                 6                    [1, 1, 1, 1, 3, 3]
        E@FG                 6                    [1, 1, 1, 2, 2, 3]
        E@HW                 6                    [0, 1, 2, 2, 2, 3]
        E@N?                 6                    [1, 1, 2, 2, 2, 2]
        E@Ow                 6                    [0, 1, 2, 2, 2, 3]
        E@Q?                 6                    [1, 1, 1, 1, 1, 1]
        E@QW                 6                    [1, 1, 1, 2, 2, 3]
        E@T_                 6                    [0, 2, 2, 2, 2, 2]
        E@YO                 6                    [1, 1, 2, 2, 2, 2]
        EG?W                 6                    [0, 1, 1, 1, 1, 2]
        EGCW                 6                    [0, 1, 1, 2, 2, 2]
        E_?w                 6                    [1, 1, 1, 1, 1, 3]
        E_Cg                 6                    [1, 1, 1, 1, 2, 2]
        E_Cw                 6                    [1, 1, 1, 2, 2, 3]
        E_Ko                 6                    [1, 1, 2, 2, 2, 2]
        F????                7                    [0, 0, 0, 0, 0, 0, 0]
        F???G                7                    [0, 0, 0, 0, 0, 1, 1]
        F???W                7                    [0, 0, 0, 0, 1, 1, 2]
        F???w                7                    [0, 0, 0, 1, 1, 1, 3]
        F??@w                7                    [0, 0, 1, 1, 1, 1, 4]
        F??Bw                7                    [0, 1, 1, 1, 1, 1, 5]
        F??GW                7                    [0, 0, 0, 0, 2, 2, 2]
        F??G_                7                    [0, 0, 0, 1, 1, 1, 1]
        F??Gg                7                    [0, 0, 0, 1, 1, 2, 2]
        F??Gw                7                    [0, 0, 0, 1, 2, 2, 3]
        F??Hg                7                    [0, 0, 1, 1, 1, 2, 3]
        F??Hw                7                    [0, 0, 1, 1, 2, 2, 4]
        F??Jg                7                    [0, 1, 1, 1, 1, 2, 4]
        F??Wo                7                    [0, 0, 0, 2, 2, 2, 2]
        F??Ww                7                    [0, 0, 0, 2, 2, 3, 3]
        F??XO                7                    [0, 0, 1, 1, 2, 2, 2]
        F??XW                7                    [0, 0, 1, 1, 2, 3, 3]
        F??Z?                7                    [0, 1, 1, 1, 1, 2, 2]
        F??ZG                7                    [0, 1, 1, 1, 1, 3, 3]
        F??^?                7                    [1, 1, 1, 1, 1, 2, 3]
        F?CJG                7                    [0, 1, 1, 1, 2, 2, 3]
        F?CPW                7                    [0, 0, 1, 2, 2, 2, 3]
        F?CZ?                7                    [0, 1, 1, 2, 2, 2, 2]
        F?C_w                7                    [0, 0, 1, 2, 2, 2, 3]
        F?Ca?                7                    [0, 1, 1, 1, 1, 1, 1]
        F?CaW                7                    [0, 1, 1, 1, 2, 2, 3]
        F?Ch_                7                    [0, 0, 2, 2, 2, 2, 2]
        F?CqO                7                    [0, 1, 1, 2, 2, 2, 2]
        F?LCG                7                    [1, 1, 1, 1, 2, 2, 2]
        F@??W                7                    [0, 0, 1, 1, 1, 1, 2]
        F@?GW                7                    [0, 0, 1, 1, 2, 2, 2]
        FG??w                7                    [0, 1, 1, 1, 1, 1, 3]
        FG?Gg                7                    [0, 1, 1, 1, 1, 2, 2]
        FG?Gw                7                    [0, 1, 1, 1, 2, 2, 3]
        FG?Wo                7                    [0, 1, 1, 2, 2, 2, 2]
        FK??W                7                    [1, 1, 1, 1, 1, 1, 2]
        FK?GW                7                    [1, 1, 1, 1, 2, 2, 2]
        F_?@w                7                    [1, 1, 1, 1, 1, 1, 4]
        F_?Hg                7                    [1, 1, 1, 1, 1, 2, 3]
        F_?XO                7                    [1, 1, 1, 1, 2, 2, 2]

class `sage.graphs.graph_database.``GraphQuery`(graph_db=None, query_dict=None, display_cols=None, **kwds)

  Bases: `sage.graphs.graph_database.GenericGraphQuery`

  A query for an instance of GraphDatabase. This class nicely wraps the SQLQuery class located in sage.databases.database.py to make the query constraints intuitive and with as many pre-definitions as possible. (i.e.: since it has to be a GraphDatabase, we already know the table structure and types; and since it is immutable, we can treat these as a guarantee).

  Note

  SQLQuery functions are available for GraphQuery. See sage.dataabases.database.py for more details.

  INPUT:

  - `graph_db` - The GraphDatabase instance to apply the query to. (If None, then a new instance is created).
  - `query_dict` - A dictionary specifying the query itself. Format is: ‘table_name’: ‘tblname’, ‘display_cols’: [‘col1’, ‘col2’], ‘expression’:[col, operator, value] If not None, query_dict will take precedence over all other arguments.
  - `display_cols` - A list of column names (strings) to display in the result when running or showing a query.
  - `kwds` - The columns of the database are all keywords. For a database table/column structure dictionary, call graph_db_info. Keywords accept both single values and lists of length 2. The list allows the user to specify an expression other than equality. Valid expressions are strings, and for numeric values (i.e. Reals and Integers) are: ‘=’,’‘,’‘,’=’,’=’. String values also accept ‘regexp’ as an expression argument. The only keyword exception to this format is induced_subgraphs, which accepts one of the following options: 1. [‘one_of’,String,...,String] Will search for graphs containing a subgraph isomorphic to any of the graph6 strings in the list. 2. [‘all_of’,String,...,String] Will search for graphs containing a subgraph isomorphic to each of the graph6 strings in the list.

  EXAMPLES:

      sage: Q = GraphQuery(display_cols=['graph6','num_vertices','degree_sequence'],num_edges=['<=',5],min_degree=1)
      sage: Q.number_of()
      35
      sage: Q.show()
      Graph6               Num Vertices         Degree Sequence
      ------------------------------------------------------------
      A_                   2                    [1, 1]
      BW                   3                    [1, 1, 2]
      CF                   4                    [1, 1, 1, 3]
      CK                   4                    [1, 1, 1, 1]
      CL                   4                    [1, 1, 2, 2]
      CN                   4                    [1, 2, 2, 3]
      D?{                  5                    [1, 1, 1, 1, 4]
      D@s                  5                    [1, 1, 1, 2, 3]
      D@{                  5                    [1, 1, 2, 2, 4]
      DBg                  5                    [1, 1, 2, 2, 2]
      DBk                  5                    [1, 1, 2, 3, 3]
      DIk                  5                    [1, 2, 2, 2, 3]
      DK[                  5                    [1, 2, 2, 2, 3]
      D_K                  5                    [1, 1, 1, 1, 2]
      D`K                  5                    [1, 1, 2, 2, 2]
      E?Bw                 6                    [1, 1, 1, 1, 1, 5]
      E?Fg                 6                    [1, 1, 1, 1, 2, 4]
      E?N?                 6                    [1, 1, 1, 1, 2, 2]
      E?NG                 6                    [1, 1, 1, 1, 3, 3]
      E@FG                 6                    [1, 1, 1, 2, 2, 3]
      E@N?                 6                    [1, 1, 2, 2, 2, 2]
      E@Q?                 6                    [1, 1, 1, 1, 1, 1]
      E@QW                 6                    [1, 1, 1, 2, 2, 3]
      E@YO                 6                    [1, 1, 2, 2, 2, 2]
      E_?w                 6                    [1, 1, 1, 1, 1, 3]
      E_Cg                 6                    [1, 1, 1, 1, 2, 2]
      E_Cw                 6                    [1, 1, 1, 2, 2, 3]
      E_Ko                 6                    [1, 1, 2, 2, 2, 2]
      F??^?                7                    [1, 1, 1, 1, 1, 2, 3]
      F?LCG                7                    [1, 1, 1, 1, 2, 2, 2]
      FK??W                7                    [1, 1, 1, 1, 1, 1, 2]
      FK?GW                7                    [1, 1, 1, 1, 2, 2, 2]
      F_?@w                7                    [1, 1, 1, 1, 1, 1, 4]
      F_?Hg                7                    [1, 1, 1, 1, 1, 2, 3]
      F_?XO                7                    [1, 1, 1, 1, 2, 2, 2]

  `get_graphs_list`()

    Returns a list of Sage Graph objects that satisfy the query.

    EXAMPLES:

        sage: Q = GraphQuery(display_cols=['graph6','num_vertices','degree_sequence'],num_edges=['<=',5],min_degree=1)
        sage: L = Q.get_graphs_list()
        sage: L[0]
        Graph on 2 vertices
        sage: len(L)
        35

  `number_of`()

    Returns the number of graphs in the database that satisfy the query.

    EXAMPLES:

        sage: Q = GraphQuery(display_cols=['graph6','num_vertices','degree_sequence'],num_edges=['<=',5],min_degree=1)
        sage: Q.number_of()
        35

  `query_iterator`()

    Returns an iterator over the results list of the GraphQuery.

    EXAMPLE:

        sage: Q = GraphQuery(display_cols=['graph6'],num_vertices=7, diameter=5)
        sage: for g in Q:
        ...     print g.graph6_string()
        F?`po
        F?gqg
        F@?]O
        F@OKg
        F@R@o
        FA_pW
        FEOhW
        FGC{o
        FIAHo
        sage: Q = GraphQuery(display_cols=['graph6'],num_vertices=7, diameter=5)
        sage: it = iter(Q)
        sage: while True:
        ...     try: print it.next().graph6_string()
        ...     except StopIteration: break
        F?`po
        F?gqg
        F@?]O
        F@OKg
        F@R@o
        FA_pW
        FEOhW
        FGC{o
        FIAHo

  `show`(max_field_size=20, with_picture=False)

    Displays the results of a query in table format.

    INPUT:

    - `max_field_size` - width of fields in command prompt version
    - `with_picture` - whether or not to display results with a picture of the graph (available only in the notebook)

    EXAMPLES:

        sage: G = GraphDatabase()
        sage: Q = GraphQuery(G, display_cols=['graph6','num_vertices','aut_grp_size'], num_vertices=4, aut_grp_size=4)
        sage: Q.show()
        Graph6               Num Vertices         Aut Grp Size
        ------------------------------------------------------------
        C@                   4                    4
        C^                   4                    4

        sage: R = GraphQuery(G, display_cols=['graph6','num_vertices','degree_sequence'], num_vertices=4)
        sage: R.show()
        Graph6               Num Vertices         Degree Sequence
        ------------------------------------------------------------
        C?                   4                    [0, 0, 0, 0]
        C@                   4                    [0, 0, 1, 1]
        CB                   4                    [0, 1, 1, 2]
        CF                   4                    [1, 1, 1, 3]
        CJ                   4                    [0, 2, 2, 2]
        CK                   4                    [1, 1, 1, 1]
        CL                   4                    [1, 1, 2, 2]
        CN                   4                    [1, 2, 2, 3]
        C]                   4                    [2, 2, 2, 2]
        C^                   4                    [2, 2, 3, 3]
        C~                   4                    [3, 3, 3, 3]

    Show the pictures (in notebook mode only):

        sage: S = GraphQuery(G, display_cols=['graph6','aut_grp_size'], num_vertices=4)
        sage: S.show(with_picture=True)
        Traceback (most recent call last):
        ...
        NotImplementedError: Cannot display plot on command line.

    Note that pictures can be turned off:

        sage: S.show(with_picture=False)
        Graph6               Aut Grp Size
        ----------------------------------------
        C?                   24
        C@                   4
        CB                   2
        CF                   6
        CJ                   6
        CK                   8
        CL                   2
        CN                   2
        C]                   8
        C^                   4
        C~                   24

    Show your own query (note that the output is not reformatted for generic queries):

        sage: (GenericGraphQuery('select degree_sequence from degrees where max_degree=2 and min_degree >= 1',G)).show()
        degree_sequence
        --------------------
        211
        222
        2211
        2222
        21111
        22211
        22211
        22222
        221111
        221111
        222211
        222211
        222211
        222222
        222222
        2111111
        2221111
        2221111
        2221111
        2222211
        2222211
        2222211
        2222211
        2222222
        2222222

`sage.graphs.graph_database.``data_to_degseq`(data, graph6=None)

  Takes the database integer data type (one digit per vertex representing its degree, sorted high to low) and converts it to degree sequence list. The graph6 identifier is required for all graphs with no edges, so that the correct number of zeros will be returned.

  EXAMPLE:

      sage: from sage.graphs.graph_database import data_to_degseq
      sage: data_to_degseq(3221)
      [1, 2, 2, 3]
      sage: data_to_degseq(0,'D??')
      [0, 0, 0, 0, 0]

`sage.graphs.graph_database.``degseq_to_data`(degree_sequence)

  Takes a degree sequence list (of Integers) and converts to a sorted (max-min) integer data type, as used for faster access in the underlying database.

  EXAMPLE:

      sage: from sage.graphs.graph_database import degseq_to_data
      sage: degseq_to_data([2,2,3,1])
      3221

`sage.graphs.graph_database.``graph6_to_plot`(graph6)

  Constructs a graph from a graph6 string and returns a Graphics object with arguments preset for show function.

  EXAMPLE:

      sage: from sage.graphs.graph_database import graph6_to_plot
      sage: type(graph6_to_plot('D??'))
      <class 'sage.plot.graphics.Graphics'>

`sage.graphs.graph_database.``graph_db_info`(tablename=None)

  Returns a dictionary of allowed table and column names.

  INPUT:

  - `tablename` - restricts the output to a single table

  EXAMPLE:

      sage: graph_db_info().keys()
      ['graph_data', 'degrees', 'spectrum', 'misc', 'aut_grp']

      sage: graph_db_info(tablename='graph_data')
      ['complement_graph6',
       'eulerian',
       'graph6',
       'lovasz_number',
       'num_cycles',
       'num_edges',
       'num_hamiltonian_cycles',
       'num_vertices',
       'perfect',
       'planar']

`sage.graphs.graph_database.``subgraphs_to_query`(subgraphs, db)

  Constructs and returns a GraphQuery object respecting the special input required for the induced_subgraphs parameter. This input can be an individual graph6 string (in which case it is evaluated without the use of this method) or a list of strings. In the latter case, the list should be of one of the following two formats: 1. [‘one_of’,String,...,String] Will search for graphs containing a subgraph isomorphic to any of the graph6 strings in the list. 2. [‘all_of’,String,...,String] Will search for graphs containing a subgraph isomorphic to each of the graph6 strings in the list.

  This is a helper method called by the GraphQuery constructor to handle this special format. This method should not be used on its own because it doesn’t set any display columns in the query string, causing a failure to fetch the data when run.

  EXAMPLE:

      sage: from sage.graphs.graph_database import subgraphs_to_query
      sage: gd = GraphDatabase()
      sage: q = subgraphs_to_query(['all_of','A?','B?','C?'],gd)
      sage: q.get_query_string()
      'SELECT ,,,,,  FROM misc WHERE ( ( misc.induced_subgraphs regexp ? ) AND (
      misc.induced_subgraphs regexp ? ) ) AND ( misc.induced_subgraphs regexp ? )'

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