Grassmann Graphs Research Articles

A characterization of Grassmann graphs

Let q be a prime power and suppose that e and n are integers satisfying 1 ⩽ e ⩽ n − 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 F q , 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, 1 2 n, 1 2 (n ± 1) and ( e ≠ 1 2 (n±2) if q ϵ {2, 3}) and (e ≠ 1 2 (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.

A Unified Approach to a Characterization of Grassmann Graphs and Bilinear Forms Graphs

Wilbrink and Brouwer [18] proved that certain semi-partial geometries with some weak restrictions on parameters satisfy the dual of Pasch's axiom. Inspired by their work, a class of incidence structures associated with distance-regular graphs with classical parameters is studied in this paper. As a consequence, the Grassman graphs and the bilinear forms graphs are characterized simultaneously among distance-regular graphs with classical parameters, together with some extra geometric conditions.

A characterization of Grassmann and Johnson graphs

Grassmann graphs and Johnson graphs are graphs in which the neighbor of no vertex contains 3-claws, and the intersection of the neighbor of two vertices at distance 2 is edge regular. We characterize Grassmann graphs, Johnson graphs, a quotient of the Johnson graph, Schläfli graphs, and Gosset graphs by the above local properties and the additional conditions, without the hypotheses of parameters.