Contractibility and the clique graph operator

To any graph G we can associate a simplicial complex ∆(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever ∆(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whose clique graph is a cone is contractible.