Study Of The Spectral Graph Energy And Its Related Problems

Spectral graph theory is one of the important branch in graph theory, and the study of graph energy is a hot research topic in recent years. The graph energies are represented by graph spectrum which have important applications in the fied of computer science, physics,chemistry, biology and control engineering. The graph spectrum is closely related to the struct of a graph. Let G = (V(G),E(G)) be a simple undirected graph with vertex set V(G) and edge set E(G) . The adjacency matrix of graph G is denoted by A(G) . The Laplacian matrix of graph G is denoted by L(G) . The signless Laplacian matrix of graph G is denoted by Q(G) . Their eigenvalues constitute adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. The study object of graph energy is all kinds of matrices and their spectrum.In this paper, some classes of graphs are studied: the corona-vertex of the subdivision graph G1(?)G2, the corona-edge of the subdivision graph G1?G2 generated by G1 and G2,generalized R-vertex corona graph R(G)(?)?inHi constructed by G and H1,H2,...,Hn,Archimedean Duals Lattices, et al. The generalized characteristic polynomial of the graph is obtained by the generalized matrix of the graph, and then the A - spectrum, L - spectrum,Q - spectrum are obtained, which solve the problem of getting the spectrum of these complex graphs. As applications, We calculate the number of spanning trees, Kirchhoff index and energy of the graphs. We propose a method to compute spectral energy by computer programming and get the some energy indices of Archimedes Duals lattice for limited vertices.The main results are as follows:(1) Calculate and prove the generalized characteristic polynomial of the generalized R-vertex corona graph; give the number of spanning trees and the Kirchhoff index in some special cases; construct a class of generalized cospectral graph.(2) Design method and write computer programming, the exact energy values of three kinds of spectral energies of the Archimedes Duals lattice are obtained.(3) The generalized characteristic polynomial of the corona-vertex of the subdivision graph G1?G2, the corona-edge of the subdivision graph G1?G2 are obtained for larger range of graphs.(4) Calculate the numerical range of the Estrada index by computer and find its almost energy graph. At the same time, count the distance and index of the HOMO-LUMO.