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Given a graph $G$, the Motzkin and Straus formulation of the maximum clique problem is the quadratic program (QP) formed from the adjacent matrix of the graph $G$ over the standard simplex. It is well-known that the global optimum value of this QP (called Lagrangian) corresponds to the clique number of a graph. It is useful in practice if similar results hold for hypergraphs. In this paper, we attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques when the number of edges is in a certain range. Specifically, we obtain upper bounds for the Lagrangian of a hypergraph when the number of edges is in a certain range. These results further support a conjecture introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). We also establish an upper bound of the clique number in terms of Lagrangians for hypergraphs.
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참고문헌 (27)
참고문헌 신청
I. M. Bomze / 1997 / Evolution towards the maximum clique / J. Global Optim 10(2):143~164
M. Budinich / 2003 / Exact bounds on the order of the maximum clique of a graph / Discrete Appl. Math 127(3):535~543
S. Busygin / 2006 / A new trust region technique for the maximum weight clique problem / Discrete Appl. Math 154(15):2080~2096
A. Chakraborty / 2013 / Dynamics on and of complex networks. Vol. 2, Model. Simul. Sci. Eng. Technol / Birkhauser/Springer :201~220
P. Frankl / 1989 / Extremal problems whose solutions are the blowups of the small Witt-designs / J. Combin. Theory Ser. A 52(1):129~147
M. Budinich / 2003 / Exact bounds on the order of the maximum clique of a graph / Discrete Appl. Math 127(3):535~543
S. Busygin / 2006 / A new trust region technique for the maximum weight clique problem / Discrete Appl. Math 154(15):2080~2096
A. Chakraborty / 2013 / Dynamics on and of complex networks. Vol. 2, Model. Simul. Sci. Eng. Technol / Birkhauser/Springer :201~220
P. Frankl / 1989 / Extremal problems whose solutions are the blowups of the small Witt-designs / J. Combin. Theory Ser. A 52(1):129~147
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