Abstract
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in Can J Math 17:533---540 (1965). This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin---Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin---Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in Comb Probab Comput 11:199---216 (2002) and a result in Peng and Zhao (Graphs Comb, 29:681---694, 2013).
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