Study On A_α-Spectral Radius Of Unicyclic Graph

Algebraic graph theory is an important part of discrete mathematics,which mainly uses some algebraic methods to solve problems in graph theory.Graph theory is an important branch of graph theory,which mainly studies the properties of graphs and some characteristics of their structures.In order to study graph theory,people introduced the concept of graph matrix,such as adjacency matrix,Laplace matrix,unsigned Laplace matrix and distance matrix.Sometimes,the combination parameters of graphs,such as chromatic number,matching number,connectivity and maximum or minimum degree,are used to describe some results of graphs.The spectrum of graphs can be divided into Aα-spectrum,adjacency spectrum,Laplace spectrum and so on.Spectrogram theory is widely used in computer,physics,chemistry and communication network.At present,the Aα-spectral radius of graphs has attracted extensive attention of scholars in many branches of spectrum theory and become a research hotspot.This thesis mainly studies the Aα-spectral radius of unicyclic graphs.Suppose G=(V,E)is composed of n=|V| vertices and m=|E| edges.For arbitrary α∈[0,1],Nikiforov gives the definition of the Aα matrix of graph G:Aα(G)=αD(G)+(1-α)A(G),Where A(G)is the adjacency matrix of G and D(G)is the degree diagonal matrix of G.The maximum eigenvalue of a matrix Aα(G)is called the Aα-spectral radius of a graph,and is denoted as λαG).This thesis is mainly divided into three parts.The first part mainly introduces the research background and some basic concepts,and lists the research status and achievements of Aα-spectral radius at home and abroad.The second part mainly lists some lemmas and inferences that will be used in the following proof,and illustrates these lemmas and inferences more directly with pictures.The third part is the key chapter of this paper,which mainly proves the Aα-spectral radius of the first three unicyclic graphs with fixed matching number,perfect matching and the number of k hanging points,and finally summarizes the main theorems of this paper.Figure[11]Table[0]Reference[70]...