Integral Formulas For Graph Energies And Estrada Indices

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph.In 1940,Coulson obtained a well-known integral formula,so-called Coulson integral formula,which gives a relationship between the energy and the characteristic polynomial of a graph.Meanwhile,Coulson integral formula provides approaches for comparing the values of the energies of two graphs with same order.Zhou proposed the concept of the sum of powers of the Laplacian eigenvalues of graphs,and we call it the general Laplacian energy-like invariant of graphs.We propose the concept of the general energy of graphs as a generalization of graph energy,and introduce the concepts of the general energy and the Estrada index of polynomials.In this thesis,we give some integral formulas for the general energy,the general Laplacian energy-like invariant and the Estrada index of graphs by using the theories of complex integration and complex polynomials.Furthermore,we give some integral formulas for the general energy and the Estrada index of polynomials.In Chapter 1,we introduce some basic concepts and the background of the topics.Meanwhile,we give a brief introduction to the research contents,main results in this thesis.In Chapter 2,we give an introduction to some basic concepts and results from complex analysis,including Fundamental Theorem of Algebra,Cauchy's Theorem,Cauchy Integral Formula,Jordan's Lemma and the theory of residue of functions.In Chapter 3,we obtain some integral formulas for the general energy of graphs in the case that the exponent is a rational number.We discuss the case that the exponent is an irrational number.We also obtain integral formulas for the general energy of polynomials with real roots in the case that the exponent is a rational number.Furthermore,we study the general energy of trees and the integral formulas for the 2l-th spectral moment of graphs where l is a positive integer.In Chapter 4,we obtain some integral formulas for the(complex)energy of(complex)polynomials.We also obtain integral formulas for the real part and imaginary part of the(complex)energy of(complex)polynomials,respectively.In Chapter 5,we obtain some integral formulas for the general Laplacian energy-like invariant of graphs in the case that the exponent is a rational number.We discuss the case that the exponent is an irrational number.We also obtain integral formulas for the general energy of polynomials with non-negative real roots in the case that the exponent is a rational number.In Chapter 6,we obtain some integral formulas for the Estrada index of graphs and polynomials,respectively.Meanwhile,we obtain some integral formulas for the sum of sine of the eigenvalues and the sum of cosine of the eigenvalues of a graph,respectively.In Chapter 7,we briefly summarize the results obtained in the thesis and propose several problems for further study.