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# Graph—Undirected graphs with self loops

## Overview

class Graph(incoming_graph_data=None, **attr)

  Base class for undirected graphs.

  A Graph stores nodes and edges with optional data, or attributes.

  Graphs hold undirected edges. Self loops are allowed but multiple (parallel) edges are not.

  Nodes can be arbitrary (hashable) Python objects with optional key/value attributes, except that `None` is not allowed as a node.

  Edges are represented as links between nodes with optional key/value attributes.

  Parameters:
    incoming_graph_data input graph (optional, default: None)

      Data to initialize graph. If None (default) an empty graph is created. The data can be any format that is supported by the to_networkx_graph() function, currently including edge list, dict of dicts, dict of lists, NetworkX graph, 2D NumPy array, SciPy sparse matrix, or PyGraphviz graph.

    attr keyword arguments, optional (default= no attributes)

      Attributes to add to graph as key=value pairs.

  See also

  `DiGraph`
  `MultiGraph`
  `MultiDiGraph`

  Examples

  Create an empty graph structure (a “null graph”) with no nodes and no edges.

      >>> G = nx.Graph()

  G can be grown in several ways.

  Nodes:

  Add one node at a time:

      >>> G.add_node(1)

  Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).

      >>> G.add_nodes_from([2, 3])
      >>> G.add_nodes_from(range(100, 110))
      >>> H = nx.path_graph(10)
      >>> G.add_nodes_from(H)

  In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph.

      >>> G.add_node(H)

  Edges:

  G can also be grown by adding edges.

  Add one edge,

      >>> G.add_edge(1, 2)

  a list of edges,

      >>> G.add_edges_from([(1, 2), (1, 3)])

  or a collection of edges,

      >>> G.add_edges_from(H.edges)

  If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when adding nodes or edges that already exist.

  Attributes:

  Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively.

      >>> G = nx.Graph(day="Friday")
      >>> G.graph
      {'day': 'Friday'}

  Add node attributes using add_node(), add_nodes_from() or G.nodes

      >>> G.add_node(1, time="5pm")
      >>> G.add_nodes_from([3], time="2pm")
      >>> G.nodes[1]
      {'time': '5pm'}
      >>> G.nodes[1]["room"] = 714  # node must exist already to use G.nodes
      >>> del G.nodes[1]["room"]  # remove attribute
      >>> list(G.nodes(data=True))
      [(1, {'time': '5pm'}), (3, {'time': '2pm'})]

  Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edges.

      >>> G.add_edge(1, 2, weight=4.7)
      >>> G.add_edges_from([(3, 4), (4, 5)], color="red")
      >>> G.add_edges_from([(1, 2, {"color": "blue"}), (2, 3, {"weight": 8})])
      >>> G[1][2]["weight"] = 4.7
      >>> G.edges[1, 2]["weight"] = 4

  Warning: we protect the graph data structure by making `G.edges` a read-only dict-like structure. However, you can assign to attributes in e.g. `G.edges[1, 2]`. Thus, use 2 sets of brackets to add/change data attributes: `G.edges[1, 2]['weight'] = 4` (For multigraphs: `MG.edges[u, v, key][name] = value`).

  Shortcuts:

  Many common graph features allow python syntax to speed reporting.

      >>> 1 in G  # check if node in graph
      True
      >>> [n for n in G if n < 3]  # iterate through nodes
      [1, 2]
      >>> len(G)  # number of nodes in graph
      5

  Often the best way to traverse all edges of a graph is via the neighbors. The neighbors are reported as an adjacency-dict `G.adj` or `G.adjacency()`

      >>> for n, nbrsdict in G.adjacency():
      ...     for nbr, eattr in nbrsdict.items():
      ...         if "weight" in eattr:
      ...             # Do something useful with the edges
      ...             pass

  But the edges() method is often more convenient:

      >>> for u, v, weight in G.edges.data("weight"):
      ...     if weight is not None:
      ...         # Do something useful with the edges
      ...         pass

  Reporting:

  Simple graph information is obtained using object-attributes and methods. Reporting typically provides views instead of containers to reduce memory usage. The views update as the graph is updated similarly to dict-views. The objects `nodes`, `edges` and `adj` provide access to data attributes via lookup (e.g. `nodes[n]`, `edges[u, v]`, `adj[u][v]`) and iteration (e.g. `nodes.items()`, `nodes.data('color')`, `nodes.data('color', default='blue')` and similarly for `edges`) Views exist for `nodes`, `edges`, `neighbors()`/`adj` and `degree`.

  For details on these and other miscellaneous methods, see below.

  Subclasses (Advanced):

  The Graph class uses a dict-of-dict-of-dict data structure. The outer dict (node_dict) holds adjacency information keyed by node. The next dict (adjlist_dict) represents the adjacency information and holds edge data keyed by neighbor. The inner dict (edge_attr_dict) represents the edge data and holds edge attribute values keyed by attribute names.

  Each of these three dicts can be replaced in a subclass by a user defined dict-like object. In general, the dict-like features should be maintained but extra features can be added. To replace one of the dicts create a new graph class by changing the class(!) variable holding the factory for that dict-like structure.

  node_dict_factoryfunction, (default: dict)

    Factory function to be used to create the dict containing node attributes, keyed by node id. It should require no arguments and return a dict-like object

  node_attr_dict_factory: function, (default: dict)

    Factory function to be used to create the node attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object

  adjlist_outer_dict_factoryfunction, (default: dict)

    Factory function to be used to create the outer-most dict in the data structure that holds adjacency info keyed by node. It should require no arguments and return a dict-like object.

  adjlist_inner_dict_factoryfunction, (default: dict)

    Factory function to be used to create the adjacency list dict which holds edge data keyed by neighbor. It should require no arguments and return a dict-like object

  edge_attr_dict_factoryfunction, (default: dict)

    Factory function to be used to create the edge attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object.

  graph_attr_dict_factoryfunction, (default: dict)

    Factory function to be used to create the graph attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object.

  Typically, if your extension doesn’t impact the data structure all methods will inherit without issue except: `to_directed/to_undirected`. By default these methods create a DiGraph/Graph class and you probably want them to create your extension of a DiGraph/Graph. To facilitate this we define two class variables that you can set in your subclass.

  to_directed_classcallable, (default: DiGraph or MultiDiGraph)

    Class to create a new graph structure in the `to_directed` method. If `None`, a NetworkX class (DiGraph or MultiDiGraph) is used.

  to_undirected_classcallable, (default: Graph or MultiGraph)

    Class to create a new graph structure in the `to_undirected` method. If `None`, a NetworkX class (Graph or MultiGraph) is used.

  Subclassing Example

  Create a low memory graph class that effectively disallows edge attributes by using a single attribute dict for all edges. This reduces the memory used, but you lose edge attributes.

      >>> class ThinGraph(nx.Graph):
      ...     all_edge_dict = {"weight": 1}
      ...
      ...     def single_edge_dict(self):
      ...         return self.all_edge_dict
      ...
      ...     edge_attr_dict_factory = single_edge_dict
      >>> G = ThinGraph()
      >>> G.add_edge(2, 1)
      >>> G[2][1]
      {'weight': 1}
      >>> G.add_edge(2, 2)
      >>> G[2][1] is G[2][2]
      True

## Methods

### Adding and removing nodes and edges

`Graph.__init__`([incoming_graph_data]) | Initialize a graph with edges, name, or graph attributes.
`Graph.add_node`(node_for_adding, **attr) | Add a single node `node_for_adding` and update node attributes.
`Graph.add_nodes_from`(nodes_for_adding, **attr) | Add multiple nodes.
`Graph.remove_node`(n) | Remove node n.
`Graph.remove_nodes_from`(nodes) | Remove multiple nodes.
`Graph.add_edge`(u_of_edge, v_of_edge, **attr) | Add an edge between u and v.
`Graph.add_edges_from`(ebunch_to_add, **attr) | Add all the edges in ebunch_to_add.
`Graph.add_weighted_edges_from`(ebunch_to_add) | Add weighted edges in `ebunch_to_add` with specified weight attr
`Graph.remove_edge`(u, v) | Remove the edge between u and v.
`Graph.remove_edges_from`(ebunch) | Remove all edges specified in ebunch.
`Graph.update`([edges, nodes]) | Update the graph using nodes/edges/graphs as input.
`Graph.clear`() | Remove all nodes and edges from the graph.
`Graph.clear_edges`() | Remove all edges from the graph without altering nodes.

### Reporting nodes edges and neighbors

`Graph.nodes` | A NodeView of the Graph as G.nodes or G.nodes().
`Graph.__iter__`() | Iterate over the nodes.
`Graph.has_node`(n) | Returns True if the graph contains the node n.
`Graph.__contains__`(n) | Returns True if n is a node, False otherwise.
`Graph.edges` | An EdgeView of the Graph as G.edges or G.edges().
`Graph.has_edge`(u, v) | Returns True if the edge (u, v) is in the graph.
`Graph.get_edge_data`(u, v[, default]) | Returns the attribute dictionary associated with edge (u, v).
`Graph.neighbors`(n) | Returns an iterator over all neighbors of node n.
`Graph.adj` | Graph adjacency object holding the neighbors of each node.
`Graph.__getitem__`(n) | Returns a dict of neighbors of node n.
`Graph.adjacency`() | Returns an iterator over (node, adjacency dict) tuples for all nodes.
`Graph.nbunch_iter`([nbunch]) | Returns an iterator over nodes contained in nbunch that are also in the graph.

### Counting nodes edges and neighbors

`Graph.order`() | Returns the number of nodes in the graph.
`Graph.number_of_nodes`() | Returns the number of nodes in the graph.
`Graph.__len__`() | Returns the number of nodes in the graph.
`Graph.degree` | A DegreeView for the Graph as G.degree or G.degree().
`Graph.size`([weight]) | Returns the number of edges or total of all edge weights.
`Graph.number_of_edges`([u, v]) | Returns the number of edges between two nodes.

### Making copies and subgraphs

`Graph.copy`([as_view]) | Returns a copy of the graph.
`Graph.to_undirected`([as_view]) | Returns an undirected copy of the graph.
`Graph.to_directed`([as_view]) | Returns a directed representation of the graph.
`Graph.subgraph`(nodes) | Returns a SubGraph view of the subgraph induced on `nodes`.
`Graph.edge_subgraph`(edges) | Returns the subgraph induced by the specified edges.