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arXiv · 1508.03971

On Clique Convergences of Graphs

S. M. Hegde·V. V. P. R. V. B. Suresh Dara
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Abstract

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 $Q_i \cap Q_j \neq \emptyset$. 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 $G=G_1 \Box G_2$, then $K^2(G) = G$.

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S. M. Hegde, V. V. P. R. V. B. Suresh Dara. 2015-08-17. On Clique Convergences of Graphs. https://arxiv.org/abs/1508.03971

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