Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

# networkx.algorithms.bipartite.projection.projected_graph¶

`projected_graph`(B, nodes, multigraph=False)[source]

Returns the projection of B onto one of its node sets.

Returns the graph G that is the projection of the bipartite graph B onto the specified nodes. They retain their attributes and are connected in G if they have a common neighbor in B.

Parameters
BNetworkX graph

The input graph should be bipartite.

nodeslist or iterable

Nodes to project onto (the “bottom” nodes).

multigraph: bool (default=False)

If True return a multigraph where the multiple edges represent multiple shared neighbors. They edge key in the multigraph is assigned to the label of the neighbor.

Returns
GraphNetworkX graph or multigraph

A graph that is the projection onto the given nodes.

Notes

No attempt is made to verify that the input graph B is bipartite. Returns a simple graph that is the projection of the bipartite graph B onto the set of nodes given in list nodes. If multigraph=True then a multigraph is returned with an edge for every shared neighbor.

Directed graphs are allowed as input. The output will also then be a directed graph with edges if there is a directed path between the nodes.

The graph and node properties are (shallow) copied to the projected graph.

See `bipartite documentation` for further details on how bipartite graphs are handled in NetworkX.

Examples

```>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(4)
>>> G = bipartite.projected_graph(B, [1, 3])
>>> list(G)
[1, 3]
>>> list(G.edges())
[(1, 3)]
```

If nodes `a`, and `b` are connected through both nodes 1 and 2 then building a multigraph results in two edges in the projection onto [`a`, `b`]:

```>>> B = nx.Graph()
>>> B.add_edges_from([("a", 1), ("b", 1), ("a", 2), ("b", 2)])
>>> G = bipartite.projected_graph(B, ["a", "b"], multigraph=True)
>>> print([sorted((u, v)) for u, v in G.edges()])
[['a', 'b'], ['a', 'b']]
```