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Exact bounds on the order of the maximum clique of a graph. (English) Zbl 1051.68112

Summary: The paper reviews some of the existing exact bounds to the maximum clique of a graph and successively presents a new upper and a new lower bound. The new upper bound is ωn – rank A ¯/2 , where A ¯ is the adjacency matrix of the complementary graph, and derives from a formulation of the maximum clique problem in complex space. The new lower bound is ω1/(1-g j * (α * )) (see text for details) and improves strictly the present best lower bound published by H. S. Wilf [J. Comb. Theory, Ser. B 40, 113–117 (1986; Zbl 0598.05047)].

Throughout the paper an eye is kept on the computational complexity of actually calculating the bounds. At the end, the various bounds are compared on 700 random graphs.

68R10Graph theory in connection with computer science (including graph drawing)
05C50Graphs and linear algebra
05C69Dominating sets, independent sets, cliques