Distances and diameters on iterated clique graphs

  • Authors:

  • Miguel A. Pizaña

  • Affiliations:

  • Departamento de Ingeniería Elétrica, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México D.F. 09340, Mexico

  • Venue:
  • Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
  • Year:
  • 2004

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Abstract

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: K0(G)=G and Kn(G)=K(Kn-1(G)). We study the relationship between distances in G and distances in Kn(G). Then we apply these results to Johnson graphs to give a shorter and simpler proof of Bornstein and Szwarefiter's theorem: For each n there exists a graph G such that diam(Kn(G))=diam(G)+n. In the way, a new family of graphs with increasing diameters under the clique operator is shown.