Spectra Of Subdivision Vertex-edge Corona For Graphs

Spectral graph theory is one of the important ?elds of graph theory, which mainly concerned with the spectrum of a graph, and the former originated with the quantum chemistry. The main work of the spectral graph theory is to study the spectra of multiform matrices of graphs and discuss the relations between the spectra and graph invariants,structure properties of graphs by means of the matrix theory,combinatorics theory. The mathematical monograph ”Spcktren Endlicher Grafen”(1957) of L. Collatz and U. Sinogowitz is usually considered as the starting point of the study of spectral graph theory.In the past 50 years, The adjacently spectral graph theory has been developed into a research hotspot of algebraic graph theory and extensively to many natural science ?elds.In 1973, E. H¨uckel introduced the H¨uckel molecular orbital theory, which established the relationship between the molecular orbital theory levels and the spectrum of the molecular graph. In addition, the Laplacian spectrum of graph is the most extensive ?eld of spectral graph theory research. The Laplacian matrix of graph, denoted by Kirchhoff matrix, can be traced back to the Matrix-tree Theorem found by the Kirchhoff when he studied on current theory. Because of this well-known theorem, it closely contacts the spanning trees of a graph with the Laplacian matrix together. Research on the Laplacian spectrum of graph not only has important theoretical value but also has been widely application in chemistry, physics, complex network and other ?elds. Thus, this impulses the development of the spectral graph theory greatly.In this paper, we introduce a graph operation on G1, G2 and G3and the result graph is denoted by GS1?(GV2∪ GE3). This thesis focus on spectrum of GS1?(GV2∪ GE3) and cospectral graphs of GS1?(GV2∪ GE3). This thesis is organized as follows:In chapter 1, ?rstly we simply introduce the research background of the spectrum theory. Secondly we give some notations. Thirdly we summarize the background and overseas and domestic research status of spectra of graph obtained by graph operation.Finally, we outline the main results of this thesis.In chapter 2, the adjacency spectra, the Laplacian spectra and signless Laplacian spectra of GS1?(GV2∪ GE3) are determined whenever G1 is regular. As applications,we construct in?nitely many pairs of A-cospectral graphs, L-cospectral graphs and Qcospectral graphs. In addition, the number of spanning trees of GS1?(GV2∪ GE3) is also determined.