Research On The A_? Spectral Radius Of Graphs

Spectral graph theory is a research hotspot of algebraic graph theory.In the research process,people have introduced many matrices closely related to the structure of graphs,such as:adjacency matrix,signless laplacian matrix,etc.Spectral graph theory is the research that whether or not the properties of the graphs can be reflected by the algebraic properties of these matrices(mainly the eigenvalues of these matrices).The spectrum of the graph can be mainly divided into adjacent spectra,signless laplacian spectra,etc.It is not difficult to find in the research that the spectral radius and signless laplacian spectral radius have similar properties.Therefore,we can combine the two when considering the problem.Let G be a graph of order n with adjacency matrix A(G)and let D(G)be the diagonal matrix of the degrees of G.For any real ? ?[0,1],write A?a(G)for the matrix A?(G)=?D(G)+(1-?)A(G).Moreover,the spectral radius of Aa(G)is called the Aa spectral radius of G,written as Pa(G).The contributions of this paper are as follows:·In Chapter 1,we introduce the background and significance of this paper briefly.And we give the basic concepts and symbols involved in this paper.It is no difficult to find that the adjacent spectral radius and the signless Laplacian spectral radius have similar results and these results can often be generalized to the Aa spectral radius.In this chapter,we introduce these results.Moreover,we study the structure of the graphs with specific forbidden subgraph on ?(G),q(G)and Aa(G).·In Chapter 2,we study the structure of the outerplanar graph with maximum ??(G).And we determine the upper bound on ??(G)if ? ?(0,1)and G is a graph with no K2,t minor.In this chapter,we prove that when ??(0,1)and n?:48(1-?)2/?3(5+4(1-?)/(?))+1 the unique outerplanar graph of order n with maximum pa(G)is the join of a vertex and a path Pn-1.This result generalizes previous result on ?(G)due to Michael Tait and Josh Tobin[14]to pa(G).And we prove ??(G)? ??(F3(n))if G is a graph with no K2,3 minor.Moreover,when ? ?(0,1),t? 4 and n? 16(1-?)2/?3(5+4(1-?)(?))t+1,we prove ??(G)?t+?n-1/2+(?)if G is a graph with no K2,t minor.These results generalize previous results on ?(G)due to Nikiforov[33]to pa(G).·In Chapter 3,we consider the structure of the planar graph with maximum ??(G).We just consider ??[0.486,1/2]in this chapter.Firstly,we prove ??(G)>?(n+2)if G is a planar graph of order n(n is sufficiently large).Secondly,we prove that there exist two vertexes of degree n-1 for such graph G.Finally,we show that the unique planar graph of order n with maximum ??(G)is the join of an edge and a path Pn-2.Obviously,the unique planar graph of order n with maximum signless Laplacian spectral radius is the graph K2?Pn-2.These results generalize previous results on ?(G)due to Michael Tait and Josh Tobin[14]to pa(G).·In Chapter 4,we describe the structure of the unicyclic graph with maximum pa(G)with fixed diameter.Firstly,we prove that for any G ? und,3 ?d ?n-2,the graph with maximum ??(G)in und is ?nd or ?nd.Secondly,by the above conclusion,we prove that the the graph with maximum poa(G)in und is the graph ?nd or the graph ?nd.These results generalize previous results on ?(G)and q(G)due to Liu Huiqing,Lu Mei[49]and He Shushan,Li Shuchao[50]to ??(G).