The Spectra Of Some Lattice Graphs On Surfaces

The spectral graph theory is one of the important fields of graph theory, which mainly concerned with the spectrum and Laplacian spectrum of a graph, and the formeroriginated from the quantum chemistry. In 1931, E. Hiickel introduced the Huckel molecular orbital theory, which established the relationship between the molecular orbital energy levels and the spectrum of the molecular graph, and impulse the developmentof the spectral graph theory greatly. 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 "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 hotspot of algebraic graph theory and applied extensively to many natural science fields.In this thesis, we consider the spectra of some lattice graphs on surfaces( polyhexes,4-8 lattice graph and 4-6-8 lattice graph on torus, Klein bottle and cylinder) which have some backgrounds in chemistry and physics. And by the quotient spectralgraph theory, we obtain the relations between their spectra. Toroidal polyhexes, Klein bottle polyhexes and Cylinder polyhexes are the graphs of trivalent hexagonal tessellation of corresponding surfaces, and 4-8 (4-6-8) lattice graphs on the three surfacesare also edge-to-edge tilings with two (three) types of polygons—quadrangle and octagon (quadrangle, hexagon and octagon).There are five chapters altogether in the thesis. In the first chapter, besides introducingthe background of the spectral graph theory , some concepts and notations, we mainly present the research problems and their developments, and introduce the main results obtained in the following chapters.In chapter 2, we discuss the spectra of toroidal polyhexes which are determined by a string (p,q,t) and thus denoted by H(p,q,t) (p≥1,q≥1,0≤t≤p - 1). By the theory of block circulant matrices we obtain the explicit expression for eigenvaluesof H(p,q, 0),λ=±(?),for l = 0,1,…,p - 1and k = 0,1,…,q - 1. Then the asymptotic behavior of the energy of H(p,q,0) is obtained by the technique of double integral, that is, the energy (the sum of the absolute values of the eigenvalues) of H(p, q, 0) grows as 9.8935pq, as p, q→∞. Moreover, we prove that, 0 is an eigenvalue of H(p, 2, t) if and only if p and (t - 2) can be both divided by 3.In chapter 3, we mainly discuss the relation between spectra of toroidal polyhexes H(p,q,t) and Klein bottle Polyhexes which are determined by a string (p, q) and denoted by K(p,q) (p≥1,q≥1). According to the common quotient of H(p,q,t)and K(p, q), 2q common eigenvaluesλ=±(?)(k = 0,1,…, q - 1) of themare determined. In addition, we show that their quotient relationship: according toγis odd or even, K(p,q) is the quotient of K(p,γq) or H(p,γq,p -γ/2q), respectively.In chapter 4, we discuss the spectra of zig-zag open-ended nanotubes T(p, q) and armchair open-ended nanotubes T_A(p,q). By quotient theory, we obtain the bound of the spectral radius of T(p,q): (?) >ρ(T(p,q)) > (?) and the explicit expression for T_A(p,q):ρ(T_A(p,q)) =2 cosÏ€/(q+3)+ 1, whereρ(G) denote the spectral radius of G. In addition, the nullity of T(p, q) is calculated, that is, the nullity of T(p, q) is zero or 2(q + 1), according to p is odd or even. Finally, we show that T(p,q) and T_A(p,q) are quotients of T(γp,q) and T_A(γp, q), respectively.In the last chapter, we obtain the factorizations of the characteristic polynomials of toroidal 4-8 lattice graph HG₁(p, q, 0) and 4-6-8 lattice graph HG₂(p, q, 0), whose factors are quartic polynomials and six degree polynomials, respectively. Then we determine the nullity of HG₂(p,q,0): if p is odd, its nullity equals to q + 1 and if p is even, its nullity is 2q + 3 or 2q + 1, according to q divided by 4 or not. Finally, we discuss some quotient relations of them.