Isometric embeddings of Johnson graphs in Grassmann graphs

Pankov, Mark

Journal of Algebraic Combinatorics, Tome 33 (2011) no. 4, pp. 555-570.

Résumé

Summary: Let $V$ be an $n$-dimensional vector space ($4\leq n\infty $) and let $G _{ k}( V)$ mathcalG_k$(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma _{ k }( V)$. We describe all isometric embeddings of Johnson graphs $J( l, m), 1 m l - 1$ in $\Gamma _{ k }( V), 1 k n - 1$ (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of $J( n, k)$ in $\Gamma _{ k }( V)$ is an apartment of $G _{ k}( V)$ mathcalG_k$(V)$ if and only if $n=2 k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in $\Gamma _{ k }( V), 1 k n - 1$.

Détail

Keywords: keywords Johnson graph, Grassmann graph, building, apartment