On Clique Convergences of Graphs

TL;DR

This paper investigates the properties of clique graphs, especially their convergence, in various graph operations like joins and Cartesian products, providing formulas, conditions for completeness, and convergence criteria.

Contribution

It offers new characterizations of clique convergence and completeness conditions for clique graphs in join and Cartesian product graphs.

Findings

Number of cliques in clique graphs of joins is determined.

Necessary and sufficient conditions for clique graph completeness are established.

Clique convergence in graph joins is characterized, with specific results for Cartesian products.

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 $K^2(G) = G$.