Upper Bound Estimation Of Lagrangians For A Class Of Uniform Hypergraphs

Lagrangians of hypergraph occupy the important position in the graph theory, atthe same time, he also has many applications in real life. For example, it has playedan important role in optimal control, mathematical model and biochemistry research.Therefore, it is of great significance in the study of Lagrangians of hypergraph.In1941, Tur′an [30] found that for given p and n, V=[n], the maximal quantity ofthe edge of the graph without p-clique satisfiesThis is the famous Tura′n theorem. It connects Lagrangians with clique. Tur′an’s theoremis an important result in graph theory, and it has the profound influence. In1965,Motzkin and Straus [21] found a simple and efective method solving Lagrangians ofgraph. This method has been applied in many optimization problems. Their resultstates that the Lagrangian of a graph is the same as the Lagrangian of its maximumcliques. This result not only provided a new proof Tur′an’s theorem but also arousedinterests in the study of Lagrangians of hypergraphs. The following idea is popularizingthe method on hypergraphs. However, the obvious generalization of Motzkin-Straus’sresults are not exactly. There are many examples of hypergraphs whose Lagrangian isnot the same as the Lagrangian of any its proper subgraphs. The result of Motzkizand Straus is not completely wrong either. Their results are still correct in certainconditions. For example in [25], let m and t be positive integers satisfying G be a3-graph with m edges and contain a clique of order t1. ThenThough Motzkin and Straus found the method to solve Lagrangians of graphs andgave a further proof of Tura′n’s theorem, limitations of result on hypergraphs hinderedthe further development of research. However, sometimes we don’t need to know aboutLagrangian of hypergraph. We only need its upper bound. This paper shows my resultbased on the fixed number of vertices.Next we define a class of uniform hypergraphs. For a given set of vertices V,We divide the vertice set into r disjoint parts {Vi1}1≤i1≤r, and then We divide everyVi1(1≤i1≤r) into r disjoint parts {Vi1i2}1≤i2≤r. And then We divide every Vi1i2(1≤i1, i2≤r) into r disjoint parts {Vi1i2i3}1≤i3≤r. Repeat the above operation until we getrpdisjoint sets {Vi1i2···ip}1≤i1, i2,···, ip≤r. The edge of this class of hypergraphs includesonly one vertice of each Vi, or includes only one vertice of each Vi1i2, until includes onlyone vertice of each Vi1i2···ip. And take all possible edges. We define this hypergraph asS(r)p[V]. We speculate that the Lagrangian of this class hypergraphs is the upper boundwhen the vertice set is divided as symmetrically as possible. In this paper,we show thatin3-uniform hypergraphs.