  There is a remarkable connection between the maximum clique number and the
Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This
connection and its extensions were successfully employed in optimization to
provide heuristics for the maximum clique number in graphs. It is useful in
practice if similar results hold for hypergraphs. In this paper, we explore
evidences that the Lagrangian of a 3-uniform hypergraph is related to the order
of its maximum cliques when the number of edges of the hypergraph is in certain
range. In particular, we present some results about a conjecture introduced by
Y. Peng and C. Zhao (2012) and describe a combinatorial algorithm that can be
used to check the validity of the conjecture.
