On clique convergence of graphs

Abstract

Let G be a graph and K<inf>G</inf> be the set of all cliques of G, then the clique graph of G denoted by K(G) is the graph with vertex set K<inf>G</inf> and two elements Q<inf>i</inf>,Q<inf>j</inf>?K<inf>G</inf> form an edge if and only if Q<inf>i</inf>?Q<inf>j</inf>?0?. Iterated clique graphs are defined by K0(G)=G, and Kn(G)=K(Kn?1(G)) for n>0. In this paper we prove a necessary and sufficient condition for a clique graph K(G) to be complete when G=G<inf>1</inf>+G<inf>2</inf>, give a partial characterization for clique divergence of the join of graphs and prove that if G<inf>1</inf>, G<inf>2</inf> are Clique-Helly graphs different from K<inf>1</inf> and G=G<inf>1</inf>?G<inf>2</inf>, then K2(G)=G. © 2016 Kalasalingam University