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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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