We consider a class of graphs which satisfies a set of certain conditions. Let G, G' be such graphs with set of vertices P, P'. Let 1 <= k < dima(G) be a positive integer, and let phi : P -> P' be a surjection satisfying d(x, y) <= k double left right arrow d(x(phi), y(phi)) <= k for all x, y is an element of P. We show that phi is an isomorphism between G and G'. This result is applied to the graphs arising from the adjacency relations of the spaces of rectangular matrices, symmetric matrices. Hermitian matrices, alternate matrices, and Grassmann spaces. (C) 2010 Elsevier Inc. All rights reserved.