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.
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Pankov, M. Isometric embeddings of Johnson graphs in Grassmann graphs. J Algebr Comb 33, 555–570 (2011). https://doi.org/10.1007/s10801-010-0258-0
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DOI: https://doi.org/10.1007/s10801-010-0258-0