Grassmann graph

Grassmann graph
Named after	Hermann Grassmann
Vertices	${\binom {n}{k}}_{q}$
Edges	${\frac {q[k]_{q}[n-k]_{q}}{2}}{\binom {n}{k}}_{q}$
Diameter	min(k, n – k)
Properties	Distance-transitive Connected
Notation	Jq(n,k)
Table of graphs and parameters

In graph theory, Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph J_q(n, k) are the k-dimensional subspaces of an n-dimensional vector space over a finite field of order q; two vertices are adjacent when their intersection is (k – 1)-dimensional.

Many of the parameters of Grassmann graphs are q-analogs of the parameters of Johnson graphs, and Grassmann graphs have several of the same graph properties as Johnson graphs.

Graph-theoretic properties

• J_q(n, k) is isomorphic to J_q(n, n – k).
• For all 0 ≤ d ≤ diam(J_q(n,k)), the intersection of any pair of vertices at distance d is (k – d)-dimensional.
• The clique number of J_q(n,k) is given by an expression in terms its least and greatest eigenvalues λ_min and λ_max:

$\omega \left(J_{q}(n,k)\right)=1-{\frac {\lambda _{\max }}{\lambda _{\min }}}$

Automorphism group

There is a distance-transitive subgroup of $\operatorname {Aut} (J_{q}(n,k))$ isomorphic to the projective linear group $\operatorname {P\Gamma L} (n,q)$.

In fact, unless $n=2k$ or $k\in \{1,n-1\}$, $\operatorname {Aut} (J_{q}(n,k))$ ≅ $\operatorname {P\Gamma L} (n,q)$; otherwise $\operatorname {Aut} (J_{q}(n,k))$ ≅ $\operatorname {P\Gamma L} (n,q)\times C_{2}$ or $\operatorname {Aut} (J_{q}(n,k))$ ≅ $\operatorname {Sym} ([n]_{q})$ respectively.

Intersection array

As a consequence of being distance-transitive, $J_{q}(n,k)$ is also distance-regular. Letting $d$ denote its diameter, the intersection array of $J_{q}(n,k)$ is given by $\left\{b_{0},\ldots ,b_{d-1};c_{1},\ldots c_{d}\right\}$ where:

• $b_{j}:=q^{2j+1}[k-j]_{q}[n-k-j]_{q}$ for all $0\leq j<d$.
• $c_{j}:=([j]_{q})^{2}$ for all $0<j\leq d$.

Spectrum

• The characteristic polynomial of $J_{q}(n,k)$ is given by

$\varphi (x):=\prod \limits _{j=0}^{\operatorname {diam} (J_{q}(n,k))}\left(x-\left(q^{j+1}[k-j]_{q}[n-k-j]_{q}-[j]_{q}\right)\right)^{\left({\binom {n}{j}}_{q}-{\binom {n}{j-1}}_{q}\right)}$.

See also

• Grassmannian
• Johnson graph