Isometric embeddings of Johnson graphs in Grassmann graphs

Abstract

Let $V$ be an $n$-dimensional vector space ($4\le n<\infty$) and let ${\mathcal{G}}_k(V)$ 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 ${\mathcal{G}}_k(V)$ if and only if $n=2k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in ${\Gamma }_k(V)$, $1<k<n-1$.

New version -- 14 pages accepted to Journal of Algebraic Combinatorics