Isometric embeddings of Johnson graphs in Grassmann graphs

TL;DR

This paper characterizes all isometric embeddings of Johnson graphs into Grassmann graphs, revealing conditions under which these embeddings form apartments and classifying rigid embeddings, advancing understanding of geometric structures in vector spaces.

Contribution

It provides a complete description of isometric Johnson graph embeddings in Grassmann graphs and classifies rigid embeddings, extending geometric and combinatorial knowledge.

Findings

All isometric embeddings of Johnson graphs are described.

Embeddings of J(n,k) are apartments iff n=2k.

Rigid embeddings are classified.

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 $\Gamma_{k}(V)$, $1<k<n-1$.