Johnson graph

Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph formula_1 are the formula_2-element subsets of an formula_3-element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains formula_4-elements. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson.
Automorphism group.
There is a distance-transitive subgroup of formula_24 isomorphic to formula_25. In fact, formula_26, unless formula_27; otherwise, formula_28.
Intersection array.
As a consequence of being distance-transitive, formula_1 is also distance-regular. Letting formula_30 denote its diameter, the intersection array of formula_1 is given by 
where:
It turns out that unless formula_1 is formula_35, its intersection array is not shared with any other distinct distance-regular graph; the intersection array of formula_35 is shared with three other distance-regular graphs that are not Johnson graphs.
Johnson scheme.
The Johnson graph formula_1 is closely related to the Johnson scheme, an association scheme in which each pair of -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 formula_2-element subsets of an formula_43-element set and whose edges correspond to disjoint pairs of subsets.
Open Problems.
The vertex-expansion properties of Johnson graphs, as well as the structure of the corresponding extremal sets of vertices of a given size, are not fully understood. However, an asymptotically tight lower-bound on expansion of large sets of vertices was recently obtained.
In general, determining the chromatic number of a Johnson graph is an open problem.