A_α-Spectral Characterization Of Several Types Of Graphs

The graph spectra theory is an important part of algebraic graph theory,which has very important applications in other related subjects.The spectral extremal problem is a hot issue in the research of the graph spectra theory in recent years,and scholars are more interested in studying the extreme graph corresponding to the maximum or minimum spectral radius of graph.Many scholars first studied the properties of adjacency matrix and Laplacian matrix of graphs,and later many scholars began to explore the properties of signless Laplacian matrix of graphs.In 2017,after Nikiforov put forward the concept of A_α-matrix,scholars have shifted their attention to study a series of related properties of graphs by using A_α-matrix of graphs.Whenα-?2,the signless Laplacian matrix is a special case of A_α-matrix.In this dissertation,the A_α-spectral extremal problems of several kinds of graphs discussed by analyzing the structure of graphs.The main contents are as follows:In Chapter 1,firstly we introduce the research background and functions of this dissertation,then gives relevant concepts and symbols involved in this dissertation,and finally,this dissertation briefly introduces the research status and thoughts.In Chapter 2,we study the A_α-spectral characterization of unicycle graphs.Firstly,the case where the A_α-spectral radius reaches the extreme value is studied by using the edge perturbation of the A_α-spectral of a graph.Secondly,the range of A_α-separator of a unicyclic graph is studied,and the corresponding extreme graph when the A_α-separator reaches the maximum described.In Chapter 3,the spectral extremal problem of Hamiltonicity is studied by using the structural properties of graphs.Firstly,the graph is densified by the closure operation of the graph,so that the structure of the complementary graph becomes relatively simple,and it is easy to find the relationship between the degree of the graph and the spectral parameters.Then,the sufficient condition of signless Laplacian spectral for the graph with larger and smallest degree has Hamiltonicity is obtained.Finally,the Hamiltonicity of the graph characterized by A_α-spectral radius is investigated.In Chapter 4,we summarize the main content of this dissertation and look forward to the further research in the future.