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:G (graph) – A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.
Returns:L – The line graph of G.
Return type:graph

Examples

>>> import networkx as nx
>>> 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)]]

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.