Turán number of Berge linear forests in uniform hypergraphs

Let F be a graph and H be a hypergraph.We say that H contains a Berge-F If there exists a bijectionψ:E(F)→E(H)such that for Ve E E(F),e C(e),and the Turan number of Berge-F is defined to be the maximum number of edges in an r-uniform hypergraph of order n that is Berge-F-free,denoted by ex,(n,Berge-F).A linear forest is a graph whose connected components are all paths or isolated vertices.Let Ln,k be the family of all linear forests of n vertices with k edges.In this paper,Turan number of Berge-Ln,in an r-uniform hypergraph is studied.When r≥k+1 and 3≤r≤l[]=1,we determine 2 the exact value of ex,(n,Berge-Ln,)respectively.When K-1≤r≤k,we 2 determine the upper bound of ex,(n,Berge-Ln,).

Uniform Hypergraphs under Certain Intersection Constraints between Hyperedges

A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to L.In this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m and L.We will provide some tight upper and lower bounds on k in terms of n,m and L.

Spectral radius of uniform hypergraphs and degree sequences

We present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

The Maximum and Minimum Degree of the Random r-uniform r-partite Hypergraphs

In this paper we consider the random r-uniform r-partite hypergraph model H（n1, n2,…, nr; n, p） which consists of all the r-uniform r-partite hypergraphs with vertex partition {V1,V2,…, Vr} where ｜Vi｜ = ni = ni（n） （1 ≤ i ≤ r） are positive integer-valued functions on n with n1 ＋n2 ＋… ＋nr =n, and each r-subset containing exactly one element in Vi （1 ≤ i ≤ r） is chosen to be a hyperedge of Hp ∈H（n1,n2,…,nr;n,p） with probability p = p（n）, all choices being independent. Let △V1 = △V1 （H） and δv1 = δv1（H） be the maximum and minimum degree of vertices in V1 of H, respectively; Xd,V1 = Xd,V1 （H）, Yd,V1 = Yd,V1 （H）, Zd,V1 = Zd,V1 （H） and Zc,d,V1=Zc,d,V1 （H） be the number of vertices in V1 of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H（n1, n2,…, nr; n,p）, Xd,V1, Yd,V1, Zd,V1, and Zc,d,V1all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D（n） such that in the space H（n1, n2,…,nr; n, p）, lim n→＋∞ P（△v1 = D（n）） = 1？ What is the range of p such that a.e., Hp ∈ H（n1,n2,...,nr;n,p） has a unique vertex in V1 with degree Av1（Hp）？ Both answers are p = o（logn1/N）, where N = r ∏ i=2 ni. The corresponding problems on δv1（Hp） also are considered, and we obtained the answers are p ≤ （1＋o（1））（logn1/N） andp=o （log n1/N）, respectively.

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.

Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor.By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix,the equality cases of the bounds are completely characterized by graph theory methods.Applying these bounds to a nonnegative irreducible matrix or a connected graph(digraph),we can improve the results of L.H.You,Y.J.Shu,and P.Z.Yuan[Linear Multilinear Algebra,2017,65(1):113-128],and obtain some new or known results.Applying these bounds to a uniform hypergraph,we obtain some new results and improve some known results of X.Y.Yuan,M.Zhang,and M.Lu[Linear Algebra Appl.,2015,484:540-549].Finally,we give a characterization of a strongly connected/c-uniform directed hypergraph,and obtain some new results by applying these bounds to a uniform directed hypergraph.