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A family of clique divergent graphs with linear growth. (English) Zbl 0892.05041
Let $$k(G)$$ be the clique graph of a given graph $$G=(V,E)$$. This is the intersection graph of the set of all cliques of $$G$$. Its vertices are the cliques of $$G$$. Two different vertices of $$k(G)$$ are joined by an edge iff they have a non-empty intersection. The iterated clique graphs $$k^n(G)$$ are defined by $$k^{n+1} (G)= k(k^n(G))$$. $$G$$ is called $$k$$-divergent, if the sequence $$\{p_n(G)\}= \{p_n\}= \{| V(k^n (G)|\}$$, formed by the orders of these graphs, tends to infinity with $$n$$. It is an interesting question whether there exist graphs with polynomial growth, that means, whether there exists a $$k$$-divergent graph $$G$$ such that for some fixed polynomial $$f$$ holds $$p_n\leq f(n)$$ for all $$n$$.
In the present paper an infinite set of finite graphs $$G$$ is given such that, for any graph $$G$$, $$p_n(G)$$ is a linear function of $$n$$. Further there are given examples of graphs $$G$$ such that $$p_n(G)$$ is a polynomial of any given positive degree. Therefore, firstly powers $$C^s_n$$ of cycles $$C_n$$ are considered by the authors, and some characterizing properties of them are given. Using these graphs, the graph $$G=G(r,s)$$ of order $$rs$$ and size $$rs(s-1)+r$$, $$r,s\geq 3$$, is introduced which is obtained from $$C^{s-1}_{rs}$$ by adding some $$r$$ edges. The authors’ first result is that this graph $$G=G (r,s)$$ satisfies $$p_n(G)= r\cdot n+o(G)$$ for all $$n$$. This means that $$G$$ is a $$k$$-divergent graph with linear growth and growth rate $$r\geq 3$$. In a further paper it is also proved that the growth rate $$r=1$$ and $$r=2$$ can be realized. Concrete examples for this result are given here. Finally, by applying the above-mentioned first result in a certain product formula of the strong product of two special graphs the authors get the following second application by induction: For any integer $$d\geq 1$$, there exists a graph $$G$$ such that $$p_n(G)$$ is a polynomial of degree $$d$$ in $$n$$.

##### MSC:
 05C75 Structural characterization of families of graphs
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##### References:
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