Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
sage: g = graphs.PetersenGraph()
sage: print g.distance_all_pairs()
{0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2
Testing on Random Graphs:
sage: g = graphs.RandomGNP(20,.3)
sage: distances = g.distance_all_pairs()
sage: all([g.distance(0,v) == distances[0][v] for v in g])
True
See Also:
•distance_matrix()
distance_graph(dist)
Returns the graph on the same vertex set as the original graph but vertices are adjacent in the returned
graph if and only if they are at specified distances in the original graph.
INPUT:
•dist is a nonnegative integer or a list of nonnegative integers. Infinity may be used here to
describe vertex pairs in separate components.
OUTPUT:
The returned value is an undirected graph. The vertex set is identical to the calling graph, but edges of the
returned graph join vertices whose distance in the calling graph are present in the input dist. Loops will
only be present if distance 0 is included. If the original graph has a position dictionary specifying locations
of vertices for plotting, then this information is copied over to the distance graph. In some instances this
layout may not be the best, and might even be confusing when edges run on top of each other due to
symmetries chosen for the layout.
EXAMPLES:
sage: G = graphs.CompleteGraph(3)
sage: H = G.cartesian_product(graphs.CompleteGraph(2))
sage: K = H.distance_graph(2)
sage: K.am()
[0 0 0 1 0 1]
[0 0 1 0 1 0]
[0 1 0 0 0 1]
[1 0 0 0 1 0]
[0 1 0 1 0 0]
[1 0 1 0 0 0]
To obtain the graph where vertices are adjacent if their distance apart is d or less use a range() command
to create the input, using d+1 as the input to range. Notice that this will include distance 0 and hence
place a loop at each vertex. To avoid this, use range(1,d+1).
sage: G = graphs.OddGraph(4)
sage: d = G.diameter()
sage: n = G.num_verts()
sage: H = G.distance_graph(range(d+1))
sage: H.is_isomorphic(graphs.CompleteGraph(n))
False
sage: H = G.distance_graph(range(1,d+1))
sage: H.is_isomorphic(graphs.CompleteGraph(n))
True
1.1. Generic graphs 45