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 Paschʹs 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.