The Grassmann graph is defined such that the vertices are the -dimensional subspaces of an -dimensional finite field of order and edges correspond to pairs of vertices whose intersection is -dimensional.

has vertex count , where is a -binomial, and edge count

is isomorphic to .

The graph is related to Kirkman's schoolgirl problem.

Grassmann graphs are distance-transitive and therefore also distance-regular.

Many parameters of
are *q*-analogs of the corresponding parameters
of the Johnson graph .