The Spectral Characterization Of Special Hypergraph

Spectrum theory is a subject which is widely studied and applied in combinatorics,computer science and social science.It mainly studies the relationship between the structural and spectral properties of graphs.The study of extremal values and extremal graphs of spectral parameters are hot topics in the study of the graph theory in the recent years.Scholars are keen on studying the extremal graph corresponding to the maximum or minimum spectral radius of a graph.In 2012,Cooper and Dutle defined the eigenvalues and spectra of a uniform hypergraph as the eigenvalues and spectra of the adjacency tensor of the hypergraph,and introduced edge operation,edge release operation and other operations in the hypergraph.In this paper,we discuss the spectral extremal problem of hypergraphs by using edge operation.We characterize the extremal graph of the second largest signless Laplacian spectral radius of k uniform supertree,and determine the extremal graph of the special Berge hypergraph when its p-spectral radius reaches the maximum.The main contents are arranged as following:In Chapter 1,firstly,we introduce the development process of similar topics at home and abroad.Then,we give the relevant concepts and symbols that can be used in this paper.Finally,we recommend the research status and thoughts of this topics.In Chapter 2,we use the edge operation to study the second largest signless Laplacian spectral radius of supertree with diameter of 4.In Chapter 3,firstly,we investigate there is a 3-unifrom Berge hypergraph which all edges have a common vertex by using edge operation.Then,we character the extremal graph of 3-uniform special Berge hypergraph with the maximum p-spectral radius.In Chapter 4,we summarize the main contents of the first two chapters and discuss to the contents that can be further studied in the future.