Stars of graphs of projective codes

Let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph whose vertex set is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $\mathrm{\Pi }(n,k{)}_q$ formed by projective codes. We show that there are precisely two types of maximal cliques in $\mathrm{\Pi }(n,k{)}_q$: stars and tops. We give a complete description of stars, i.e., maximal cliques consisting of all $k$-dimensional projective codes containing a certain $(k-1)$-dimensional subspace of $V$.