Spectral Theory Of Signless Laplacian Matrix Of Hypergraph

After the concept of adjacency tensor was put forward,the tensor spectrum theory in the field of hypergraph has developed rapidly,especially in describing the spectral structure of hypergraph related tensor and the combinatorial properties of hypergraph.However,it is not easy to define the appropriate tensor and spectral book of hypergraph,so this thesis wants to study hypergraph from matrix.Then this thesis mainly studies the spectral theory of signless Laplacian matrix of hypergraph.Signless Laplacian matrix have many useful properties,such as symmetric,non-negative,semi-definite posetive and irreducible.These properties provide a lot of convenience for the study of hypergraphs.At the same time,it is also because more complex calculation and higher theoretical cost are needed when using tensor to study hypergraph.In the process of paying attention to the development of tensors in the field of hypergraphs,people also ignore the important role of matrices in the study of hypergraphs.These functions also remind that matrix is still of great help in the study of hypergraphs.Firstly,this thesis expounds the research background and main problems of this thesis.In this thesis,the structure corresponding to the maximum spectral radius and the structure corresponding to the minimum eigenvalue of hypergraph class are characterized by signless Laplacian matrix,and the symbols used in this thesis are given.Then the main research results and structure are given.The first part of this thesis is to study the connected k-uniform hypergraph with the maximum signless Laplacian spectral radius given the number of pendent vertices.The hypergraph structure under the maximum spectral radius corresponding to the signless Laplacian matrix in various cases of n-r ≥k and n-r ∈[k-1] is characterized by using the methods of characteristic equation and edge shift transformation.The second part of this thesis is to study the minimum H eigenvalue of signless Laplacian matrix of hypergraph.Since all eigenvalues of the matrix are positive,the minimum H eigenvalue is the minimum eigenvalue.In this thesis,the concepts of partially bipartite hypergraph and balanced partially bipartite hypergraph are introduced.In this thesis,some theorems for characterizing the structure of hypergraphs with minimum signless Laplacian Eigenvalues are obtained through the influence of the branching and moving of hypergraphs on eigenvalues by connecting it with the properties of partially bipartite and balanced partially bipartite of bipartite hypergraph,we obtain the class of hypergraph with minimum eigenvalue in bipartite hypergraph.Finally,this thesis summarizes and prospects the hypergraph structure studied.Compared with tensor,signless Laplacian matrix is simple to calculate and has various good properties,which is of great help to the study of hypergraph.In the following,we can also use this matrix to study the upper and lower bounds of the radius of the hypergraph.For the hypergraph with a given number of edges,the uniform hypertree of a given degree sequence can also be used.These studies are based on k-uniform hypergraphs.We can also try to extend them to general hypergraphs and directed hypergraphs.