Study On The Related Problems Of Graph Spectrum

Graph spectrum is a set of all the characteristic roots and the number of its associated matrices, it is a specific property of the specific combination structure,and can give some profound results. Graph spectrum is one of the important fields in Algebraic graph theory. Its development not only promotes and enriches graph theory and combinatorics theory and related subjects, but also has wide applications in information science, image processing, compressed sensing, quantum chemistry,physics, computer science, network, information technology and integrated circuit design. In particular, graph spectrum is closely related with many aspects in graph energy, the sum of powers of the eigenvalues of graphs, Kirchhoff index of graph, the random walk of the complex network and so on. The full text is divided into five chapters and the concrete research content is as follows:In Chapter 1, we first introduce some basic concepts, terminology and notations,then we survey the research background and developments of the graph spectrum and its related problems. Finally we give a brief induction to main results of our work.In Chapter 2, we first give the adjacency(Laplacian, signless Laplacian) characteristic polynomial and corresponding eigenvalues of the graphs with pockets. Then,we constructe two new types of graph operations: Q-vertex neighbourhood corona and Q-edge neighbourhood corona and determine the adjacency(resp.,Laplacian and signless Laplacian)spectrum. Finally, we give some sufficient and necessary conditions of graphs to be integral from which we construct infinite a new class of integral graphs.In Chapter 3, we establish some new upper and lower bounds of s*α(G) and Sα(G)for a connected graph G. The sum of power of eigenvalues of graphs is defined as theα-th power of the non-zero eigenvalues of a graph G. First, we give some new lower and upper bounds on s*α(G). As a result of these bounds, we also give some results on degree Kirchhoff index. And we obtain some new lower and upper bound on Sα(G)for G.In Chapter 4, we determine the maximal incidence energy IE(G) among all connected graphs with the connectivity κ and edge connectivity κ′. We prove that Kκ∨(K1∪ Kn-κ-1) is attained applying the relations among the Coulson integral formula, the characteristic polynomial of subdivision graph, signless Laplacian characteristic polynomial and incidence energy.In Chapter 5, we consider the resistance distance and the Kirchhoff index of some graphs. Firstly, for corona and edge corona of graph, we obtain the formulas for resistance distance and Kirchhoff index of them using the generalized inverse of the Laplacian matrix. In [117], a new graph operation: vertex-edge corona is introduced, and its A-spectra(resp., L-spectra) are investigated, but the expression of Kf((GS1?(GV2∪ GE3)) is more complicated. So we continue to give the formulas for resistance distance and Kirchhoff index of subdivision vertex-edge corona using the generalized inverse of the Laplacian matrix. We further generalize this operation which it copy the same graph extending to copy the different graph, i.e. generalized subdivision-vertex and subdivision-edge corona for graphs. We give the resistance distance and Kirchhoff index of generalized subdivision-vertex and subdivision-edge corona for graphs. Lastly, we establish the relationship between the Laplacian polynomials of G1⊙RG2and G1ΘRG2and the Laplacian polynomials of G1 and G2.Based on this, we get the formulas for Kirchhoff index of G1⊙RG2and G1ΘRG2.