The Maximal A_?-spectral Radius Of Graphs With Given Matching Number

Algebraic graph theory is an important branch of discrete mathematics.It mainly uses some algebraic methods to solve the problems in graph theory.Graph theory is one of the important branches of graph theory.It mainly studies the properties and structural characteristics of graphs.In order to study some structural properties of graphs,the concept of graph matrix is introduce.Generally,we study graph matrices such as adjacency matrix,Laplace matrix,signless Laplace matrix and distance matrix.The application of graph theory is very wide,especially in computer science,chemistry,communication network and so on.In many branches of graph theory,the study of spectral radius of graph has always been a popular topic.In this article,we mainly study the A?-spectral radius.Let G be a graph of order n with degree diagonal matrix D(G),then the A?(G)-matrix of G is defined by A?(G)=?D(G)+(1-?)A(G),where 0???1.And the largest eigenvalue of A?(G)is called the A?(G)-spectral radius of G.This paper is divided into three parts.In the first part,we introduce the research background and some basic concepts,and list the existing results for A?-spectral ra-dius and related parameters.In the second part,we list some lemmas and inferences that will be used in the following proofs and calculate the A?-spectrum of K??(?).In the third part,we first prove the properties of graphs with the largest A?-spectral radius in Gn,?,where Gn,? is the set of graphs of order n with matching number ?.Finally we prove the main theorems of this paper.