Spectral Theory Of Incidence Q-Tensors Of Hypergraphs

By using the definition of tensors,characteristic equations and properties of tensors,we study the bounds and extremum structures of several kinds of the incidence Q-spectral radius of uniform hypergraphs by using the methods of moving edges operation,edge-releasing operation and total grafting operation.In this thesis,we give some bounds of uniform hypergraphs with given degree sequence,uniform hypergraphs with given clique number and uniform hypergraphs.And we describe the extremum structures of the incidence Q-spectral radius of uniform hypergraphs given the number of pendent vertices and pendent edges,respectively;and the extremum structures of the incidence Q-spectral radius of unicyclic uniform hypergraphs and bicyclic uniform hypergraphs given the number of edges,respectively.First,we describe the main problems and research background,and give the symbols and related concepts related to hypergraphs and tensors respectively,and then give the structure of this thesis.In the first part of this thesis,we study the bounds of the incidence Q-spectral radius of uniform hypergraphs.According to the definition and properties of incidence Q-tensor,the upper bounds of the incidence Q-spectral radius of uniform hypergraphs for given degree sequence,average 2-row sum sequence are given by calculation,inference and proof.The bounds are compared by two examples.Secondly,by defining the eigenvalue and eigenvector of the tensor,the upper bound of the incidence Q-spectral radius of uniform hypergraphs is given.Finally,based on the definition and properties of clique number of hypergraphs,the bounds of the incidence Q-spectral radius of uniform hypergraphs with the given clique number are given.In the second part of this thesis,we study the extremum structure of the incidence Q-spectral radius of uniform hypergraphs.The extremum structure of the incidence Q-spectral radius of uniform hypergraphs with a given number of pendent vertices is studied,according to the properties of incidence Q-tensor of uniform hypergraphs,the extremum structure of the incidence Q-spectral radius of some uniform hypergraphs with a number of pendent vertices is characterized by its characteristic equation and moving edges operation.Then the extremum structure of the incidence Q-spectral radius of unicyclic uniform hypergraphs,bicyclic uniform hypergraphs and uniform hypergraphs under edge constraint is studied: Firstly,according to the characteristic equation of incidence Q-tensor of uniform hypergraphs and some related lemmas,the extremum structure of the incidence Q-spectral radius of uniform hypergraphs with given the number of pendent edges is characterized by calculation and moving edges operation.Secondly,according to the properties of incidence Q-tensor of uniform hypergraphs,the extremum structure of the incidence Q-spectral radius of unicyclic uniform hypergraphs with given the numbers of edges is characterized by moving edges operation.Finally,according to the properties of incidence Q-tensor of uniform hypergraphs,the extremum structure of the incidence Q-spectral radius of bicyclic uniform hypergraphs with given the numbers of edges is characterized by moving edges operation.At the end of this thesis,the results of the research are summarized,and the research content is prospected.Combined with the originality of this thesis and the research significance of the research problem,further research is needed on the spectrum problem of incidence Q-tensor of hypergraphs,and some related problems to be studied in the next step are proposed.