  A remarkable connection between the order of a maximum clique and the
Lagrangian of a graph was established by Motzkin and Straus in [7]. 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 [19] and
a result in [11].
