Journal of Algebraic Combinatorics

, Volume 33, Issue 4, pp 555–570

Isometric embeddings of Johnson graphs in Grassmann graphs

Authors

    • Department of Mathematics and InformaticsUniversity of Warmia and Mazury
Open AccessArticle

DOI: 10.1007/s10801-010-0258-0

Cite this article as:
Pankov, M. J Algebr Comb (2011) 33: 555. doi:10.1007/s10801-010-0258-0

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 Γk(V). We describe all isometric embeddings of Johnson graphs J(l,m), 1<m<l−1 in Γk(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 Γk(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 Γk(V), 1<k<n−1.

Keywords

Johnson graphGrassmann graphBuildingApartment
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© The Author(s) 2010