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Lower bounds on the number of triangles in a graph. (English) Zbl 0673.05046
Summary: For a simple graph G, let $$f(z)\equiv 1-c_ 1z+c_ 2z^ 2-c_ 3z^ 3+..$$. where $$c_ k$$ is the number of complete subgraphs on k nodes in G. Let r(G) be the reciprocal of the smallest real root of f(z). Let $$\lambda(\bar G)$$ be the spectral radius of the complement of G. We show $$r(G)\geq \lambda (\bar G)+1.$$ This is used to show that if $$c^ 2_ 1/4\leq c_ 2\leq c^ 2_ 1/3$$, then a lower bound on the number of triangles in G is $$c_ 3\geq [9c_ 2c_ 1-2c^ 3_ 1-2(c^ 2_ 1- 3c_ 2)^{3/2}]/27.$$ This improves a bound of Bollobás and is asymptotically sharp.
Also, this paper shows that $$c_ 3\leq c_ 2(\sqrt{8c_ 2+1}-3)/6$$ (a corollary of a result from Erdős and Hanini) and the average number of triangles in a graph with $$c_ 1$$ nodes and $$c_ 2$$ edges is $$E(c_ 3)=(4/3)(c^ 2_ 2-3c_ 2+2)c_ 2/(c^ 3_ 1-5c_ 1-4).$$ These are graphically compared to the best known lower bounds.

##### MSC:
 05C35 Extremal problems in graph theory
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