Distances and diameters on iterated clique graphs

If G is a graph, its clique graph, K(G), is the intersection graph of all its (maximal) cliques. Iterated clique graphs are then defined recursively by: K-o(G) = G and K-n(G) = K(Kn-1(G)). We study the relationship between distances in G and distances in K-n(G). Then we apply these results to Johnson graphs to give a shorter and simpler proof of Bornstein and Szwarcfiter's theorem: For each n there exists a graph G such that diam(K-n(G)) = diam(G) + n. In the way, a new family of graphs with increasing diameters under the clique operator is shown. (C) 2003 Elsevier B.V. All rights reserved.