Study On The Characteristic Polynomials And Spectra Of Some Graphs

The spectral graph theory is one of the important fields of graph theory,which mainly concerned with the spectrum, Laplacian spectrum and singless Laplacian spectrum of a graph. The spectrum of a graph originated from the quantum chemistry. In 1931, E. H¨uckel introduced the H¨uckel molecular orbital theory, which established the relationship between the molecular orbital energy levels and the spectrum of the molecular graph, and impulse the development of the spectral graph theory greatly. The mathematical monograph”Spektren 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 spectral graph theory has been developed into a research hot spot of algebraic graph theory and applied extensively to graph theory,physics, quantum chemistry, computer science, internet technology etc. The main work of the spectral graph theory is to study the spectral of multiform matrices of graphs and discuss the relations between the spectral and graph invariants, structure properties of graphs by means of the matrix theory, combinatorics theory. Determining the spectra of a graph is a core problem in spectral graph theory. This thesis is focus on this fundamental problem about some hexagonal graph and a rooted product graph, and discuss the relations between the spectra and graph invariants.There are five chapters altogether in the thesis. A brief organization is given in the following:In Chapter 1, we firstly introduce the background and some applications of graph spectra, then we give the notions, symbols and graphs involved in this work. Thirdly we briefly summarize the source, background of the problem.At last, we list our main results of the thesis.In Chapter 2, we firstly give the Laplacian spectra of hexacyclic chains（Fn）and M¨obius hexacyclic chains（Mn）. Secondly, we give the Laplacian energy of Fnand Mn, and determine the bound of their Laplacian energy, which is about six times of the number of hexagons.In Chapter 3, we consider three hexagonal system graphs: H_3,n, H_3,n^r, H_3,n^b.We firstly give the characteristic polynomial of H_3,nand then determine its spectral radius, the multiplicity of ±1, the number of Kek¨ul′e structure and the nullity. Secondly, we give the adjacency characteristic polynomial of H_3,n^rand then give the adjacency characteristic polynomial and Laplacian characteristic polynomial of H_3,n^b, respectively.In Chapter 4, we firstly give a factorization theorem, that is, the characteristic polynomial of GR S（σ, k） can be expressed the product of the characteristic polynomial of k weighted k-cyclic σ-digraphs. In later, we obtain the formula of characteristic polynomial for the hexagonal lattice, 8.8.4 lattice, which cylindrical boundary condition.In Chapter 5, we firstly give the adjacency spectrum of rooted product graph of a connected graph and a path, and give the eigenvectors corresponding to eigenvalue of the graph. Second, we construct the infinite families of graphs with distinct eigenvalues and controllable graphs constructed from G,respectively. At last, we discuss the Q-spectra of such graphs.