Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
AUTHOR:
Rob Beezer, 2009-11-25
distance_matrix()
Returns the distance matrix of the (strongly) connected (di)graph.
The distance matrix of a (strongly) connected (di)graph is a matrix whose rows and columns are indexed
with the vertices of the (di) graph. The intersection of a row and column contains the respective distance
between the vertices indexed at these position.
Warning: The ordering of vertices in the matrix has no reason to correspond to the order of vertices
in vertices(). In particular, if two integers i, j are vertices of a graph G with distance matrix M,
then M[i][i] is not necessarily the distance between vertices i and j.
EXAMPLES:
sage: G = graphs.CubeGraph(3)
sage: G.distance_matrix()
[0 1 1 2 1 2 2 3]
[1 0 2 1 2 1 3 2]
[1 2 0 1 2 3 1 2]
[2 1 1 0 3 2 2 1]
[1 2 2 3 0 1 1 2]
[2 1 3 2 1 0 2 1]
[2 3 1 2 1 2 0 1]
[3 2 2 1 2 1 1 0]
The well known result of Graham and Pollak states that the determinant of the distance matrix of any tree
of order n is (-1)^{n-1}(n-1)2^{n-2}
sage: all(T.distance_matrix().det() == (-1)^9*(9)*2^8 for T in graphs.trees(10))
True
See Also:
•distance_all_pairs() – computes the distance between any two vertices.
distances_distribution(G)
Returns the distances distribution of the (di)graph in a dictionary.
This method ignores all edge labels, so that the distance considered is the topological distance.
OUTPUT:
A dictionary d such that the number of pairs of vertices at distance k (if any) is equal to d[k] ·
|V (G)| · (|V (G)| − 1).
Note: We consider that two vertices that do not belong to the same connected component are at infinite
distance, and we do not take the trivial pairs of vertices (v, v) at distance 0 into account. Empty (di)graphs
and (di)graphs of order 1 have no paths and so we return the empty dictionary {}.
EXAMPLES:
An empty Graph:
sage: g = Graph()
sage: g.distances_distribution()
{}
1.1. Generic graphs 47