Research On The Fundamental Properties Of Hypergraph Extensional Eigenvalues

Graph is a tool to study the binary relation of discrete objects,and graph is an effective model to study the binary subset of finite sets.The vertices in a graph represent binary objects and edges represent the relations between binary objects.However,in real life,there is often a certain relation among multiple objects,which cannot be accurately reflected by the study of graph.Hypergraph is a natural extension of ordinary graph,which studies the relation among multiple objects.A hyperedge contains a number of vertices,which means that these vertices have the same relations.Therefore,hypergraph model can more accurately reflect the relation among complex multiple objects in reality.Several types of matrix eigenvalues of a graph can reflect structural parameters such as diameter,chromatic number and connectivity of the graph and so on.Therefore many scholars have devoted themselves to the study of matrix eigenvalues in recent years and have achieved many excellent research results.In 2021,Cheng Tao,Li hua Feng et al.first proposed the concept of extensional eigenvalues of a graph.The extensional eigenvalue of graghs is general generalization of the eigenvalue of graghs and plays an important role in characterizing the structural properties of graphs.In this paper,matrix is used as a tool to extend the extensional eigenvalues of graphs to the extensional eigenvalues of hypergraphs,and matrix and matrix are defined as the corresponding symmetric matrix and positive definite matrix of hypergraph respectively.Firstly,the bounds of the extensional eigenvalue of hypergraph are characterized,and then the bounds of the extensional spectral radius of hypergraph are described by the row sum(column sum)of matrix and matrix.Secondly,the relationship between the extensional eigenvalues of the subhypergraph and those of the original hypergraph are studied after deleting suspended edges and deleting isolated points.Finally,we study the inclusion sets of extensional eigenvalues of hypergragh(Gersgorin inclusion set and Brauer inclusion set)and compare the inclusion relation of the two inclusion sets.