Diameter preserving surjections in the geometry of matrices

Abstract: We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping $\phi:\Gamma\to\Gamma'$ 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).