Relation To Johnson Scheme
The Johnson graph is closely related to the Johnson scheme, an association scheme in which each pair of k-element sets is associated with a number, half the size of the symmetric difference of the two sets. The Johnson graph has an edge for every pair of sets at distance one in the association scheme, and the distances in the association scheme are exactly the shortest path distances in the Johnson graph.
The Johnson scheme is also related to another family of distance-transitive graphs, the odd graphs, whose vertices are -element subsets of an -element set and whose edges correspond to disjoint pairs of subsets.
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