Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping phi : Gamma -> Gamma' between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Gamma are at a distance equal to the diameter of Gamma if, and only if, their images are at a distance equal to the diameter of Gamma'. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices. and Grassmann spaces (projective spaces of rectangular matrices). (C) 2008 Elsevier Inc. All rights reserved.