Tops of graphs of projective codes

Abstract

Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. Denote by $\Pi[n,k]_q$ the subgraph of $\Gamma_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^{\Pi}_{k}$ of $\Pi[n,k]_q$ consisting of all $k$-dimensional projective codes contained in a fixed $(k+1)$-dimensional subspace of $V$. We show when and in how many lines of ${\mathcal G}_{k}(V)$ they are contained. Next we prove that $\langle U]^{\Pi}_{k}$ is a maximal clique of $\Pi[n,k]_q$ exactly if it is contained in at most one line of ${\mathcal G}_{k}(V)$.