On Clique Convergences of Graphs

Let $G$ be a graph and ${\mathcal{K}}_G$ be the set of all cliques of $G$, then the clique graph of G denoted by $K(G)$ is the graph with vertex set ${\mathcal{K}}_G$ and two elements $Q_i,Q_j\in {\mathcal{K}}_G$ form an edge if and only if $Q_i\cap Q_j\mathrm{\ne }\mathrm{\varnothing }$. Iterated clique graphs are defined by $K^0(G)=G$, and $K^n(G)=K(K^{n-1}(G))$ for $n>0$. In this paper we determine the number of cliques in $K(G)$ when $G=G_1+G_2$, prove a necessary and sufficient condition for a clique graph $K(G)$ to be complete when $G=G_1+G_2$, give a characterization for clique convergence of the join of graphs and if $G_1$, $G_2$ are Clique-Helly graphs different from $K_1$ and $G=G_1\mathrm{\square }{G}_2$, then $K^2(G)=G$.