, Volume 33, Issue 4, pp 555-570,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 17 Oct 2010

Isometric embeddings of Johnson graphs in Grassmann graphs

Abstract

Let V be an n-dimensional vector space (4≤n<∞) 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 Γ k (V). We describe all isometric embeddings of Johnson graphs J(l,m), 1<m<l−1 in Γ 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 Γ 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 Γ k (V), 1<k<n−1.