Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph are the -element subsets of an -element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains -elements. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson. Special cases is the complete graph . is the octahedral graph. is the complement graph of the Petersen graph, hence the line graph of . More generally, for all , the Johnson graph is the complement of the Kneser graph Graph-theoretic properties is isomorphic to For all , any pair of vertices at distance share elements in common. is Hamilton-connected, meaning that every pair of vertices forms the endpoints of a Hamiltonian path in the graph. In particular this means that it has a Hamiltonian cycle. It is also known that the Johnson graph is-vertex-connected. forms the graph of vertices and edges of an (n − 1)-dimensional polytope, called a hypersimplex. the clique number of is given by an expression in terms of its least and greatest eigenvalues: The chromatic number of is at most Automorphism group There is a distance-transitive subgroup of isomorphic to . In fact, , unless ; otherwise, . Intersection array As a consequence of being distance-transitive, is also distance-regular. Letting denote its diameter, the intersection array of is given by where: It turns out that unless is , its intersection array is not shared with any other distinct distance-regular graph; the intersection array of is shared with three other distance-regular graphs that are not Johnson graphs. Eigenvalues and Eigenvectors The characteristic polynomial of is given by where The eigenvectors of have an explicit description. Johnson scheme The Johnson graph 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 -element subsets of an -element set and whose edges correspond to disjoint pairs of subsets. Open Problems The vertex-expansion properties of Johnson graphs,