9.4.17. networkx.algorithms.operators.product.power¶

networkx.algorithms.operators.product.power(G, k)[source]¶

Returns the specified power of a graph.

The k`th power of a simple graph `G, denoted \(G^k\), is a graph on the same set of nodes in which two distinct nodes u and v are adjacent in \(G^k\) if and only if the shortest path distance between u and v in G is at most k.

Parameters:	G : graph A NetworkX simple graph object. k : positive integer The power to which to raise the graph G.
Returns:	NetworkX simple graph G to the power k.
Raises:	ValueError If the exponent k is not positive. NetworkXNotImplemented If G is not a simple graph.

Notes

This definition of “power graph” comes from Exercise 3.1.6 of Graph Theory by Bondy and Murty [R785].

References

[R785]

(1, 2) Bondy, U. S. R. Murty, Graph Theory. Springer, 2008.

Examples

The number of edges will never decrease when taking successive powers:

>>> G = nx.path_graph(4)
>>> list(nx.power(G, 2).edges())
[(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)]
>>> list(nx.power(G, 3).edges())
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]

The k`th power of a cycle graph on *n* nodes is the complete graph on *n* nodes, if `k is at least n // 2:

>>> G = nx.cycle_graph(5)
>>> H = nx.complete_graph(5)
>>> nx.is_isomorphic(nx.power(G, 2), H)
True
>>> G = nx.cycle_graph(8)
>>> H = nx.complete_graph(8)
>>> nx.is_isomorphic(nx.power(G, 4), H)
True