On graph-Lagrangians of hypergraphs containing dense subgraphs (Q467466)

Lagrangians of \(r\)-uniform hypergraphs, or simply \( r\)-graphs, were introduced for 2-graphs by \textit{T. S. Motzkin} and \textit{E. G. Straus} [Can. J. Math. 17, 533--540 (1965; Zbl 0129.39902)]. They established a remarkable connection between the global maxima of the Lagrangian of a graph \(G\) over the standard simplex and the clique number of \(G\). Also, they gave a new proof of \textit{P. Turán}'s theorem [Mat. Fiz. Lapok 48, 436--452 (1941; Zbl 0026.26903)]. In order to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum clique for hypergraphs when the number of edges is in certain ranges, two conjectures are proposed by \textit{Y. Peng} and \textit{C. Zhao} [Graphs Comb. 29, No. 3, 681--694 (2013; Zbl 1267.05185)]. In this paper, the authors provide upper bounds on the Lagrangian of a 3-graph containing dense subgraphs when the number of edges of 3-graphs is in certain ranges. The results presented in this paper provide substantial evidence for two conjectures which are proposed by Peng and Zhao [loc. cit.]. Also, the autors extend some results of Peng and Zhao [loc. cit.] and \textit{J. M. Talbot} [Comb. Probab. Comput. 11, No. 2, 199--216 (2002; Zbl 0998.05049)]. The authors of this paper specify that they are not able to extend the arguments in paper to verify conjectures of Peng and Zhao [loc. cit.] and a conjecture of \textit{P. Frankl} and \textit{Z. Füredi} [J. Comb. Theory, Ser. A 52, No. 1, 129--147 (1989; Zbl 0731.05030)] for more general case. When \( r \geq 4\), the computation is more complex.