Isometric embeddings of Johnson graphs in Grassmann graphs

2010/03/17 by Mark Pankov, Pankov, Mark
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1003.3329

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

arxiv created 2010/09/14 · arxiv updated 2010/09/15

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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