Exact bounds on the order of the maximum clique of a graph Original Research Article

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 ω⩽n−rank Ā/2, where Ā 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 ω⩾1/(1−gj∗(α∗)) (see text for details) and improves strictly the present best lower bound published by Wilf (J. Combin. Theory Ser. B 40 (1986) 113). 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.