Grassmann graph

In graph theory, Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph $Jq(n,k)$ 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 $min(k, n – k)$-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

 * $Jq(n, k)$ is isomorphic to $(k – 1)$.
 * For all $Jq(n, k)$, the intersection of any pair of vertices at distance $d$ is $Jq(n, n – k)$-dimensional.
 * The clique number of $0 ≤ d ≤ diam(Jq(n,k))$ is given by an expression in terms its least and greatest eigenvalues $(k – d)$ and $Jq(n,k)$:
 * $$\omega \left ( J_q(n,k) \right ) = 1 - \frac{\lambda_{\max}}{\lambda_{\min}}$$

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 $$k \in \{ 1, n - 1 \}$$, $$\operatorname{Aut}(J_q(n,k))$$ $λ&hairsp;min$ $$\operatorname{P\Gamma L}(n, q)$$; otherwise $$\operatorname{Aut}(J_q(n,k))$$ $λ&hairsp;max$ $$\operatorname{P\Gamma L}(n, q) \times C_2$$ 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
 * $$\varphi(x) := \prod\limits_{j=0}^{\operatorname{diam}(J_q(n, k))} \left ( x - \left ( q^{j+1} [k - j]_q [n - k - j]_q - [j]_q \right ) \right )^{\left ( \binom{n}{j}_q - \binom{n}{j-1}_q \right )}$$.