*(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 $\omega \u2a7dn$ – rank $\overline{A}/2$ , where $\overline{A}$ 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 $\omega \u2a7e1/(1-{g}_{{j}^{*}}\left({\alpha}^{*}\right))$ (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.