  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).
