Fixed block configuration GDDs with block size 6 and (3, r)-regular graphs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Most of the definitions used in the thesis will be defined, and we provide a basic survey of topics in graph theory and design theory pertinent to the topics studied in this thesis. In Chapter 2, we are concerned with the study of fixed block configuration group divisible designs, GDD(n; m; k; λ1; λ2). We study those GDDs in which each block has configuration (s; t), that is, GDDs in which each block has exactly s points from one of the two groups and t points from the other. Chapter 2 begins with an overview of previous results and constructions for small group size and block sizes 3, 4 and 5. Chapter 2 is largely devoted to presenting constructions and results about GDDs with two groups and block size 6. We show the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1, λ2) with fixed block configuration (3; 3). For configuration (1; 5), we give minimal or nearminimal index constructions for all group sizes n ≥ 5 except n = 10, 15, 160, or 190. For configuration (2, 4), we provide constructions for several families ofGDD(n, 2, 6; λ1, λ2)s. Chapter 3 addresses characterizing (3, r)-regular graphs. We begin with providing previous results on the well studied class of (2, r)-regular graphs and some results on the structure of large (t; r)-regular graphs. In Chapter 3, we completely characterize all (3, 1)-regular and (3, 2)-regular graphs, as well has sharpen existing bounds on the order of large (3, r)- regular graphs of a certain form for r ≥ 3. Finally, the appendix gives computational data resulting from Sage and C programs used to generate (3, 3)-regular graphs on less than 10 vertices.

Hamilton decompositions of 6-regular abelian Cayley graphs

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made: Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian. The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture: Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition. Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses. Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators. Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4. Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups.

Experimental investigation of regular fluids and nanofluids during flow boiling in a single microchannel at different heat fluxes and mass fluxes

The dissipation of high heat flux from integrated circuit chips and the maintenance of acceptable junction temperatures in high powered electronics require advanced cooling technologies. One such technology is two-phase cooling in microchannels under confined flow boiling conditions. In macroscale flow boiling bubbles will nucleate on the channel walls, grow, and depart from the surface. In microscale flow boiling bubbles can fill the channel diameter before the liquid drag force has a chance to sweep them off the channel wall. As a confined bubble elongates in a microchannel, it traps thin liquid films between the heated wall and the vapor core that are subject to large temperature gradients. The thin films evaporate rapidly, sometimes faster than the incoming mass flux can replenish bulk fluid in the microchannel. When the local vapor pressure spike exceeds the inlet pressure, it forces the upstream interface to travel back into the inlet plenum and create flow boiling instabilities. Flow boiling instabilities reduce the temperature at which critical heat flux occurs and create channel dryout. Dryout causes high surface temperatures that can destroy the electronic circuits that use two-phase micro heat exchangers for cooling. Flow boiling instability is characterized by periodic oscillation of flow regimes which induce oscillations in fluid temperature, wall temperatures, pressure drop, and mass flux. When nanofluids are used in flow boiling, the nanoparticles become deposited on the heated surface and change its thermal conductivity, roughness, capillarity, wettability, and nucleation site density. It also affects heat transfer by changing bubble departure diameter, bubble departure frequency, and the evaporation of the micro and macrolayer beneath the growing bubbles. Flow boiling was investigated in this study using degassed, deionized water, and 0.001 vol% aluminum oxide nanofluids in a single rectangular brass microchannel with a hydraulic diameter of 229 µm for one inlet fluid temperature of 63°C and two constant flow rates of 0.41 ml/min and 0.82 ml/min. The power input was adjusted for two average surface temperatures of 103°C and 119°C at each flow rate. High speed images were taken periodically for water and nanofluid flow boiling after durations of 25, 75, and 125 minutes from the start of flow. The change in regime timing revealed the effect of nanoparticle suspension and deposition on the Onset of Nucelate Boiling (ONB) and the Onset of Bubble Elongation (OBE). Cycle duration and bubble frequencies are reported for different nanofluid flow boiling durations. The addition of nanoparticles was found to stabilize bubble nucleation and growth and limit the recession rate of the upstream and downstream interfaces, mitigating the spreading of dry spots and elongating the thin film regions to increase thin film evaporation.

Benson's Theorem for Partial Geometries

In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values of α. Finally the theorem is extended to strongly regular graphs in Chapter 5. In addition we obtain expressions for the multiplicities of the eigenvalues of matrices related to the adjacency matrices of these graphs. Finally, a four lesson high school level enrichment unit is included to provide students at this level with an introduction to partial geometries, strongly regular graphs, and an opportunity to develop proof skills in this new context.