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 R

^{n}, the truncation of A by an integral matrix B is the restriction of A on the solution space of Bx = 0, denoted A

^{B}. S...[

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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 R

^{n}, the truncation of A by an integral matrix B is the restriction of A on the solution space of Bx = 0, denoted A

^{B}. Since the defining equations of A

^{B} have integral coefficients, these equations automatically reduce to the equations over Z

_{q}=Z/qZ, and we obtain arrangements A

^{B}/Z

_{q}. Let M(A

^{B}/Z

_{q}) be the set of the complement of the union of all hyperplanes in A

^{B}/Z

_{q}. We show that #M(A

^{B}/Z

_{q}) is a quasi-polynomial of positive integers q. The period can be expressed in terms of invariant factors of the submatrices of the defining matrix of A

^{B}. Write

#M(A

^{B}/Z

_{q}) = Σ

_{k=0}^{n}c

_{k}(q)q

^{k}.

We further show that, if u, v are positive integers such that u l v, then

lc

_{k}(u)l ≤ lc

_{k}(v)l for all 0 ≤ 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 K

_{m} 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 C

_{6} nor ⊝

_{3,2,5} and determined by its Laplacain spectrum, then the product G x K

_{m} is also a graph determined by its Laplacian spectrum. In addition, we find the cosepctral graphs of C

_{6} x K

_{m} and ⊝

_{3,2,5} x K

_{m}, 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

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