| This thesis studies two subjects. Chapters 1-4 study integral hyperplane arrangements. Chapters 5-8 are devoted to graphs determined by their Laplacian spectra. In Chapter 1, we introduce hyperplane arrangements, characteristic polynomials, and some well known results that we may use in later discussion. In Chapter 2, we independently prove the interlacing divisibility and some related results of invariant factors for integral matrices that we shall apply in later chapters. Chapter 3 contains our main results on integral hyperplane arrangements. We introduce an idea of integral hyperplane arrangements by a truncation method. Given an integral hyperplane arrangement A in Rn , the truncation of A by an integral matrix B is the restriction of A on the solution space of Bx = 0, denoted AB . Since the defining equations of AB have integral coefficients, these equations automatically reduce to the equations over Zq=Z/q Z , and we obtain arrangements AB/Zq . Let M( AB/Zq ) be the set of the complement of the union of all hyperplanes in AB/Zq . We show that ;We further show that, if u, v are positive integers such that u | v, then cku ≤ck v forall0≤k≤n. .;In Chapter 4, we study a special family of integral hyperplane arrangements: threshold arrangements and coordinate threshold arrangements. We obtain their explicit formulas of the quasi-polynomials for the complements of these arrangements. In Chapters 5-8, we give a Laplacian spectral characterization of the product of the complete graphs Km with trees, unicyclic graphs, and bicyclic graphs. More precisely, let G be a connected graph with at most two independent cycles. If G is neither C6 nor &THgr;3,2,5 and determined by its Laplacian spectrum, then the product G x Km is also a graph determined by its Laplacian spectrum. In addition, we find the cospectral graphs of C6 x Km and &THgr;3,2,5 x Km , where the case m = 1 is shown in Figure 5.1 and 5.2.;Keywords: Hyperplane arrangements, integral arrangements, truncated arrangements, threshold arrangements, coordinate threshold arrangements, Coxeter arrangements, quasi-polynomials, quasi-period, interlacing divisibility, Stirling numbers, cospectral graphs, Laplacian spectrum, L-DS graphs. |
