Edge coloring BIBDS and constructing MOELRs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Some historical uses and background are touched upon as well. The majority of the definitions are contained within this chapter as well. In Chapter 2 we consider the question whether one can decompose λ copies of monochromatic Kv into copies of Kk such that each copy of the Kk contains at most one edge from each Kv. This is called a proper edge coloring (Hurd, Sarvate, [29]). The majority of the content in this section is a wide variety of examples to explain the constructions used in Chapters 3 and 4. In Chapters 3 and 4 we investigate how to properly color BIBD(v, k, λ) for k = 4, and 5. Not only will there be direct constructions of relatively small BIBDs, we also prove some generalized constructions used within. In Chapter 5 we talk about an alternate solution to Chapters 3 and 4. A purely graph theoretical solution using matchings, augmenting paths, and theorems about the edgechromatic number is used to develop a theorem that than covers all possible cases. We also discuss how this method performed compared to the methods in Chapters 3 and 4. In Chapter 6, we switch topics to Latin rectangles that have the same number of symbols and an equivalent sized matrix to Latin squares. Suppose ab = n2. We define an equitable Latin rectangle as an a × b matrix on a set of n symbols where each symbol appears either [b/n] or [b/n] times in each row of the matrix and either [a/n] or [a/n] times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka × b mutually orthogonal equitable Latin rectangles as a k–MOELR(a, b; n). We show that there exists a k–MOELR(a, b; n) for all a, b, n where k is at least 3 with some exceptions.

Structure and properties of graphene nano disks (GND) with and without edge-dopants

Graphene is one of the most important materials. In this research, the structures and properties of graphene nano disks (GND) with a concentric shape were investigated by Density Functional Theory (DFT) calculations, in which the most effective DFT methods - B3lyp and Pw91pw91 were employed. It was found that there are two types of edges - Zigzag and Armchair in concentric graphene nano disks (GND). The bond length between armchair-edge carbons is much shorter than that between zigzag-edge carbons. For C24 GND that consists of 24 carbon atoms, only armchair edge with 12 atoms is formed. For a GND larger than the C24 GND, both armchair and zigzag edges co-exist. Furthermore, when the number of carbon atoms in armchair-edge are always 12, the number of zigzag-edge atoms increases with increasing the size of a GND. In addition, the stability of a GND is enhanced with increasing its size, because the ratio of edge-atoms to non-edge-atoms decreases. The size effect of a graphene nano disk on its HOMO-LUMO energy gap was evaluated. C6 and C24 GNDs possess HOMO-LUMO gaps of 1.7 and 2.1eV, respectively, indicating that they are semi-conductors. In contrast, C54 and C96 GNDs are organic metals, because their HOMO-LUMO gaps are as low as 0.3 eV. The effect of doping foreign atoms to the edges of GNDs on their structures, stabilities, and HOMO-LUMO energy gaps were also examined. When foreign atoms are attached to the edge of a GND, the original unsaturated carbon atoms become saturated. As a result, both of the C-C bonds lengths and the stability of a GND increase. Furthermore, the doping effect on the HOMO-LUMO energy gap is dependent on the type of doped atoms. The doping H, F, or OH into the edge of a GND increases its HOMO-LUMO energy gap. In contrast, a Li-doped GND has a lower HOMO-LUMO energy gap than that without doping. Therefore, Li-doping can increase the electrical conductance of a GND, whereas H, F, or OH-doping decreases its conductance.