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There is a remarkable connection between the maximum clique number and theLagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. Thisconnection and its extensions were successfully employed in optimization toprovide heuristics for the maximum clique number in graphs. It is useful inpractice if similar results hold for hypergraphs. In this paper, we exploreevidences that the Lagrangian of a 3-uniform hypergraph is related to the orderof its maximum cliques when the number of edges of the hypergraph is in certainrange. In particular, we present some results about a conjecture introduced byY. Peng and C. Zhao (2012) and describe a combinatorial algorithm that can beused to check the validity of the conjecture.

On Cliques and Lagrangians of 3-uniform Hypergraphs