Study On The Sum Of Squares Of Positive And Negative Eigenvalues And Spectral Radius Of Graphs

Graph spectra theory is a very active and important research field in graph theory.It has a wide range of applications in quantum chemistry,computer science,communication networks and other disciplines.Graph theory first explores the structure of graphs.For this reason,people introduce various types of matrices,such as adjacency matrix,incidence matrix,Laplacian matrix,etc.How to use the algebraic properties of matrices to reflect the structural properties of graphs has become an important topic in the study of graph spectra theory.Among them,the algebraic properties of matrices mainly refer to their eigenvalues and eigenvectors.This paper mainly considers the eigenvalues of the adjacency matrices of graphs,and uses algebraic methods to establish some connections between them and the structure of graphs.The main contents of this paper are as follows:In Chapter 1 we firstly introduce some important concepts and symbols used in this paper,secondly introduce the research background and current situation of this paper,and finally propose the main results.In Chapter 2 we consider the conjecture of min(s+,s-)?n-1 in a connected graph with n vertices proposed by Elphick et al.,where s+ and s-denote the sum of squares of positive eigenvalues and the sum of squares of negative eigenvalues of G,respectively.Firstly,we obtain the necessary conditions of min(s+,s-)? n-1,and then prove that the conjecture holds for the union graph Kk2?(Kk1c+Kn-k1-k2)by using the calculation method of a kind of adjacency matrix,and it is also found that the connected graph G with the maximum degree n-1 satisfies that s+?n-1.In Chapter 3 we study the spectral radii of the uni cyclic spiral graphs.For a prescribed unicyclic degree sequence ?,the graph with the largest spectral radius in the connected graphs corresponding to the degree sequence is called the unicyclic spiral graph,denoted by U?*.Firstly,we discuss the set of unicyclic spiral graphs corresponding to the degree sequence on n vertices,and use the major relationship to characterize the unicyclic spiral graphs with the maximum and minimum spectral radius for the given maximum degree.Secondly,we study the spectral radii of the unicyclic spiral graphs corresponding to a kind of unicyclic degree sequence without major relationship.Using the characteristic polynomials of the adjacency matrices of the graphs and the properties of the functions,the condition of judging the spectrum of them is obtained.The main conclusions are:1.If unicyclic degree sequences ?=(n-i,2,2,…,2,1,…,1)and ?'=(n-i-1,i+2,2,1,…,1),where n-i-1?i+2,then there exists a positive integer N,when n>N,we have ?(U?*)>?(U?'*).2.If unicyclic degree sequences ?=(n-i,2,2,…,2,1,…,1)and ?'=(n-i-1,n-i-1,…,n-i-1,dj+1,1…,1),where n-i-1<i+2,then there exists a positive integer N,when n>N,we have ?(U?*)<?(U?'*).In Chapter 4 we summarize the research content of this paper and propose further re-search directions.