Grassmann graph

Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph formula_1 are the formula_2-dimensional subspaces of an formula_3-dimensional vector space over a finite field of order formula_4; two vertices are adjacent when their intersection is formula_5-dimensional.
Many of the parameters of Grassmann graphs are formula_4-analogs of the parameters of Johnson graphs, and Grassmann graphs have several of the same graph properties as Johnson graphs.
Automorphism group.
There is a distance-transitive subgroup of formula_16 isomorphic to the projective linear group formula_17. 
In fact, unless formula_18 or formula_19, formula_20 formula_17; otherwise formula_20 formula_23or formula_20 formula_25 respectively.
Intersection array.
As a consequence of being distance-transitive, formula_13 is also distance-regular. Letting formula_27 denote its diameter, the intersection array of formula_13 is given by formula_29 where: