The Hoffman-Singleton Graph

The following set of drawings pertain to the Hoffman-Singleton graph, a.k.a., the Moore graph of degree 7 and diameter 2, the (7,5)-cage, and the strongly regular graph with parameters (50,7,0,1). Any vertex v in this graph has 7 neighbors. Each of the 7 neighbors is adjacent to 6 other vertices (accounting for all 50 vertices). Call the 7 sets of size 6 the I-sets for v. The I-sets are disjoint, lest there be 4-cycles in the graph; they are independent, lest there be triangles. Any two I-sets (for the same vertex) induce a matching. Any three I-sets induce 3 copies of a 6-cycle. The figures show the graphs induced by larger numbers of I-sets.
The subgraph induced by four I-sets (24 vertices).
The three 6-cycles that remain after removing an I-set from the 24 vertex graph.
The subgraph induced by five I-sets (30 vertices).
The 24 vertex graph as a subgraph of the 30 vertex graph.