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
![m(J_q(n,k))=1/2q[k]_q[n-k]_q(n; k)_q.](/images/equations/GrassmannGraph/NumberedEquation1.svg) |
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 
.
