# power#

power(G, k)[source]#

Returns the specified power of a graph.

The $$kth power of a simple graph :math:G$$, 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:
Ggraph

A NetworkX simple graph object.

kpositive 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 .

References


1. 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 kth 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