A characterization of Grassmann graphs

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Abstract

Let q be a prime power and suppose that e and n are integers satisfying 1 ⩽ en − 1. Then the Grassmann graph Γ(e, q, n) has as vertices the e-dimensional subspaces of a vector space of dimension n over the field Fq, where two vertices are adjacent iff they meet in a subspace of dimension e − 1. In this paper, a characterization of Γ(e, q, n) in terms of parameters is obtained provided that e ≠ 2, n − 2, 12n, 12(n ± 1) and (e ≠ 12(n±2) if q ϵ {2, 3}) and (e ≠ 12(n ± 3) if q = 3). As a consequence we can show that these Grassmann graphs are uniquely determined as distance-regular graphs by their intersection arrays.

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