On the Iterated Edge-Biclique Operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The edge-biclique graph of G, KBe(G), is the edge-intersection graph of the bicliques of G. A graph G diverges (resp. converges or is periodic) under an operator H whenever limk→∞ |V (Hk(G))| = ∞ (resp. limk→∞ Hk(G) = Hm(G) for some m or Hk(G) = Hk+s(G) for some k and s ≥ 2). The kth-iterated edge-biclique graph of G, K B e k ( G ) , is the graph obtained by applying the edge-biclique operator k successive times to G. In this paper we study the iterated edge-biclique operator KBe. In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator KBe and we propose some conjectures on the subject.