Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}\_q ^n),E)$


where: $Gr(k,\mathbb{F}\_q ^n)$ is the set of all subspace of $\mathbb{F}\_q$ of dimension k, and $E=\{ W\_1,W\_2 \in V | W\_1\cap W\_2 \neq \{0 \} \}$ 


Is there previous research about this and a standard name for this graph? I am looking mainly for properties with a focus on expander-like properties and the maximal clique of the graph.


What I found in the Internet is the Grassmannian graph which differs from the graph above (link: <http://www.win.tue.nl/~aeb/graphs/Grassmann.html>). 


Thanks



Graph of Grassmannian
These are graphs in the Grassmann scheme, and are the $q$-analogs of the Kneser graphs. A clique will be a collection of subspaces of dimension $k$, any two of which have non-trivial intersection. An obvious class of cliques arise as the set of all $k$-subspaces that contain a given 1-dimensional subspace; such cliques have size $\binom{n-1}{k-1}\_q$. In fact if $n\ge2k$ then the maximum size of a clique is
at most $\binom{n-1}{k-1}\_q$ and, if $n>2k$, any clique of this size must be the set of $k$-subspaces that contain a given 1-dimensional subspace. For more details and references, one source is <http://arxiv.org/pdf/math/0311535.pdf>.


