The Least H-eigenvalue Of Signless Laplacian Of Non-odd-bipartite Hypergraphs

Spectral hypergraph theory is an important research field of algebraic combinatorics and tenser theory,mainly studies the relation between the structural properties of hypergraphs and its represented tensors(or matrices),and determines the structural properties or parameters of hypergraph by the eigenvalues or eigenvectors of its represented tensors(or matrices).In 2012,Cooper and Dutle [3] introduced the adjacency tensor of hypergraphs,and they obtained several essential properties that similar to adjacency matrix of simple graphs.Nikiforov [19,20,21] further studied the related questions of the adjacency tensor of hypergraphs.And there are more and more researches to study the adjacency tensor of hypergraphs.Nikiforov [19] main discussed that which hypergraphs equips symmetric spectrum,and got different necessary and sufficient conditions to character the symmetry spectrum of the adjacency tensor of hypergraphs.In particular,Shao et al.[26] proved that a connected 6)uniform hypergraph has the symmetry spectrum if and only if zero is an H-eigenvalue of the signless Laplacian tensor of ,and zero is an H-eigenvalue of the signless Laplacian tensor of if and only if 6)is even and is odd-bipartite.In 2013,Xie et al.[27] introduced the signless Laplacian tensors Q()of an uniform hypergraph ,and established the relationship between the least H-eigenvalue of Q()and the edge connectivity and the edge cut of .As Q()is nonnegative,by Perron-Frobenius theorem,many results about its spectral radius are presented.Except the above work,the least H-eigenvalue of Q()receives little attention.In this paper,we focus on the relationship between least H-eigenvalue of Q()and its non-odd-bipartiteness.Let be a connected non-odd-bipartite hypergraph with even uniformity.The least H-eigenvalue of the signless Laplacian tensor of is simply called the least eigenvalue of and the corresponding H-eigenvectors are called the first eigenvectors of .The organization of the thesis is as follows:In Chapter one,we briefly introduce the investigation and development about spectrum of hypergraphs and tensors,some basic terminology and notation,and the main results in this thesis.In Chapter two,we give some numerical and structural properties about the first eigenvectors of which contains an odd-bipartite branch.And we characterize the hypergraph(s)whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph.In Chapter three,we discuss the relationship between the least H-eigenvalue of Q()and the non-odd-bipartiteness of .To achieve it,we introduce two parameters that can measure how close is to being a odd-bipartite hypergraph.We also present some upper bounds of the least eigenvalue of ,and prove that zero is the least limit point of the least H-eigenvalues of non-oddbipartite hypergraphs.