On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turán classical result on the Turán density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t . 1, then .(G) = .([t . 1](3)) provided (t.13).m.(t.13)+(t.22). They also conjectured: If G is an r-uniform graph with m edges not containing a clique of order t . 1, then .(G) < .([t . 1](r)) provided (t.1r).m.(t.1r)+(t.2r.1). It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m=(t.13)+(t.22). Regarding this conjecture, we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t . 1](3) E(G)| = p, then .(G) < .([t. 1](3)) provided m=(t.13)+(t.22) and t . 17p/2 + 11.