Contractibility and the clique graph operator

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