# line_graph#

line_graph(G, create_using=None)[source]#

Returns the line graph of the graph or digraph `G`.

The line graph of a graph `G` has a node for each edge in `G` and an edge joining those nodes if the two edges in `G` share a common node. For directed graphs, nodes are adjacent exactly when the edges they represent form a directed path of length two.

The nodes of the line graph are 2-tuples of nodes in the original graph (or 3-tuples for multigraphs, with the key of the edge as the third element).

For information about self-loops and more discussion, see the Notes section below.

Parameters:
Ggraph

A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.

create_usingNetworkX graph constructor, optional (default=nx.Graph)

Graph type to create. If graph instance, then cleared before populated.

Returns:
Lgraph

The line graph of G.

Notes

Graph, node, and edge data are not propagated to the new graph. For undirected graphs, the nodes in G must be sortable, otherwise the constructed line graph may not be correct.

Self-loops in undirected graphs

For an undirected graph `G` without multiple edges, each edge can be written as a set `{u, v}`. Its line graph `L` has the edges of `G` as its nodes. If `x` and `y` are two nodes in `L`, then `{x, y}` is an edge in `L` if and only if the intersection of `x` and `y` is nonempty. Thus, the set of all edges is determined by the set of all pairwise intersections of edges in `G`.

Trivially, every edge in G would have a nonzero intersection with itself, and so every node in `L` should have a self-loop. This is not so interesting, and the original context of line graphs was with simple graphs, which had no self-loops or multiple edges. The line graph was also meant to be a simple graph and thus, self-loops in `L` are not part of the standard definition of a line graph. In a pairwise intersection matrix, this is analogous to excluding the diagonal entries from the line graph definition.

Self-loops and multiple edges in `G` add nodes to `L` in a natural way, and do not require any fundamental changes to the definition. It might be argued that the self-loops we excluded before should now be included. However, the self-loops are still “trivial” in some sense and thus, are usually excluded.

Self-loops in directed graphs

For a directed graph `G` without multiple edges, each edge can be written as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L` if and only if the tail of `x` matches the head of `y`, for example, if ```x = (a, b)``` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`.

Due to the directed nature of the edges, it is no longer the case that every edge in `G` should have a self-loop in `L`. Now, the only time self-loops arise is if a node in `G` itself has a self-loop. So such self-loops are no longer “trivial” but instead, represent essential features of the topology of `G`. For this reason, the historical development of line digraphs is such that self-loops are included. When the graph `G` has multiple edges, once again only superficial changes are required to the definition.

References

• Harary, Frank, and Norman, Robert Z., “Some properties of line digraphs”, Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161–168.

• Hemminger, R. L.; Beineke, L. W. (1978), “Line graphs and line digraphs”, in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory, Academic Press Inc., pp. 271–305.

Examples

```>>> G = nx.star_graph(3)
>>> L = nx.line_graph(G)
>>> print(sorted(map(sorted, L.edges())))  # makes a 3-clique, K3
[[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]]
```

Edge attributes from `G` are not copied over as node attributes in `L`, but attributes can be copied manually:

```>>> G = nx.path_graph(4)
>>> G.add_edges_from((u, v, {"tot": u + v}) for u, v in G.edges)
>>> G.edges(data=True)
EdgeDataView([(0, 1, {'tot': 1}), (1, 2, {'tot': 3}), (2, 3, {'tot': 5})])
>>> H = nx.line_graph(G)
>>> H.add_nodes_from((node, G.edges[node]) for node in H)
>>> H.nodes(data=True)
NodeDataView({(0, 1): {'tot': 1}, (2, 3): {'tot': 5}, (1, 2): {'tot': 3}})
```