Grassmann Graph
The Grassmann graph J_q(n,k) is defined such that the vertices are the k-dimensional subspaces of an n-dimensional finite field of order q and edges correspond to pairs of vertices whose intersection is (k-1)-dimensional.
J_q(n,k) has vertex count [画像:(n; k)_q], where [画像:(n; k)_q] is a q-binomial, and edge count
![画像: m(J_q(n,k))=1/2q[k]_q[n-k]_q(n; k)_q.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/GrassmannGraph/NumberedEquation1.svg){kind=link}
J_q(n,k) is isomorphic to J_q(n,n-k).
The graph J_2(4,2) is related to Kirkman's schoolgirl problem.
Grassmann graphs are distance-transitive and therefore also distance-regular.
Many parameters of J_q(n,k) are q-analogs of the corresponding parameters of the Johnson graph J(n,k).
See also
Distance-Regular Graph, Distance-Transitive Graph, Johnson Graph, Kirkman's Schoolgirl ProblemExplore with Wolfram|Alpha
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References
Brouwer, A. "Grassmann Graphs." http://www.win.tue.nl/~aeb/graphs/Grassmann.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, 1989.Referenced on Wolfram|Alpha
Grassmann GraphCite this as:
Weisstein, Eric W. "Grassmann Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrassmannGraph.html