hoffman_singleton_graph

hoffman_singleton_graph#

graph_tool.collection.hoffman_singleton_graph()[source]#

Returns the Hoffman-Singleton Graph.

Returns:
gGraph

Hoffman–Singleton Graph with 50 nodes and 175 edges

Notes

The Hoffman–Singleton graph is a symmetrical undirected graph with 50 nodes and 175 edges.

All indices lie in \(\mathbb{Z} \mod 5\), that is, the integers modulo 5 [hoffman].

It is the only regular graph of vertex degree 7, diameter 2, and girth 5.

It is the unique (7,5)-cage graph and Moore graph, and contains many copies of the Petersen graph [hoffman-singleton].

Constructed from pentagon and pentagram as follows [hoffman-wiki]:

  1. Take five pentagons \(P_h\) and five pentagrams \(Q_i\) .

  2. Join vertex \(j\) of \(P_h\) to vertex \(h\times i + j\) of \(Q_i\).

References