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The maximum number of cliques in dense graphs. (English) Zbl 0569.05029
We will consider only undirected, connected graphs without loops or multiple edges. Denote the number of vertices of G by $$| G|$$. A clique of graph G is a maximal complete subgraph. The clique graph K(G) of G is the intersection graph of the cliques of G. The density w(G) is the number of vertices in the largest clique of G. A graph is called dense if w(G)$$\geq | G| /2.$$
This paper makes precise the intuitive idea that very dense graphs have fewer cliques than less dense graphs. First, it is shown that for any graph G, $$2^{| G| -w(G)}\geq | K(G)|.$$ Secondly, this bound is sharp among dense graphs, and among them only. In fact, for all integers s,t$$\geq 4$$ where $$t\geq s\geq t/2$$, there exists a graph G such that $$| G| =t$$, $$w(G)=s$$, and $$2^{t-s}=| K(G)|.$$ Call a dense graph packed if $$2^{| G| -w(G)}=| K(G)|.$$ The 2n- Neumann graph is the complement of a matching between 2n vertices. Thirdly, it is shown that any packed graph G contains an induced subgraph isomorphic to the 2[$$| G| -w(G)]$$-Neumann graph. Lastly, the clique graphs of packed graphs are characterized.

##### MSC:
 05C35 Extremal problems in graph theory
##### Keywords:
clique graph; density; dense graphs; Neumann graph; packed graphs
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