Isometric embeddings of Johnson graphs in Grassmann graphs
Mark Pankov · Journal of Algebraic Combinatorics · 2010
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