Abstract
This paper proposes three new analytical lower bounds on the clique number of a graph and compares these bounds with those previously established in the literature. Two proposed bounds are derived from the well-known Motzkin–Straus quadratic programming formulation for the maximum clique problem. Theoretical results on the comparison of various bounds are established. Computational experiments are performed on random graph models such as the Erdös-Rényi model for uniform graphs and the generalized random graph model for power-law graphs that simulate graphs with different densities and assortativity coefficients. Computational results suggest that the proposed new analytical bounds improve the existing ones on many graph instances.
Acknowledgments
The authors are grateful to Sergiy Butenko (Texas A&M University), Aaron Clauset (University of Colorado at Boulder) and Vladimir Nikiforov (University of Memphis) for their helpful comments.
Disclosure
No potential conflict of interest was reported by the authors.
Notes
1. Note that several ‘algorithmic’ bounds on 
have been derived [Citation6,Citation11,Citation19,Citation31,Citation44]; however, as in the case of such bounds on , they are out of scope of this paper. Besides, we do not consider lower bounds for special types of graphs, such as bounds on triangle-free graphs [Citation27,Citation28,Citation32,Citation45,Citation46] and others [Citation30].
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Funding
V. Boginski's research was supported in part by the Air Force Office of Scientific Research grant number FA9550-12-1-0103. This material is based upon work supported by AFRL Mathematical Modeling and Optimization Institute.
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