Cyclic automorphic graph decompositions

Chapter 1 introduces the tools and mechanics necessary for this report. Basic definitions and topics of graph theory which pertain to the report and discussion of automorphic decompositions will be covered in brief detail. An automorphic decomposition D of a graph H by a graph G is a G-decomposition of H such that the intersection of graph (D) @H. H is called the automorhpic host, and G is the automorphic divisor. We seek to find classes of graphs that are automorphic divisors, specifically ones generated cyclically. Chapter 2 discusses the previous work done mainly by Beeler. It also discusses and gives in more detail examples of automorphic decompositions of graphs. Chapter 2 also discusses labelings and their direct relation to cyclic automorphic decompositions. We show basic classes of graphs, such as cycles, that are known to have certain labelings, and show that they also are automorphic divisors. In Chapter 3, we are concerned with 2-regular graphs, in particular rCm, r copies of the m-cycle. We seek to show that rCm has a ρ-labeling, and thus is an automorphic divisor for all r and m. we discuss methods including Skolem type difference sets to create cycle systems and their correlation to automorphic decompositions. In the Appendix, we give classes of graphs known to be graceful and our java code to generate ρ-labelings on rCm.

Enumeration of inequivalent cycle decompositions

A k-cycle decomposition of order n is a partition of the edges of the complete graph on n vertices into k-cycles. In this report a backtracking algorithm is developed to count the number of inequivalent k-cycle decompositions of order n.

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.

PROPERTIES AND STRUCTURES OF Li-N BASED HYDROGEN STORAGE MATERIALS

Traditional transportation fuel, petroleum, is limited and nonrenewable, and it also causes pollutions. Hydrogen is considered one of the best alternative fuels for transportation. The key issue for using hydrogen as fuel for transportation is hydrogen storage. Lithium nitride (Li3N) is an important material which can be used for hydrogen storage. The decompositions of lithium amide (LiNH2) and lithium imide (Li2NH) are important steps for hydrogen storage in Li3N. The effect of anions (e.g. Cl-) on the decomposition of LiNH2 has never been studied. Li3N can react with LiBr to form lithium nitride bromide Li13N4Br which has been proposed as solid electrolyte for batteries. The decompositions of LiNH2 and Li2NH with and without promoter were investigated by using temperature programmed decomposition (TPD) and X-ray diffraction (XRD) techniques. It was found that the decomposition of LiNH2 produced Li2NH and NH3 via two steps: LiNH2 into a stable intermediate species (Li1.5NH1.5) and then into Li2NH. The decomposition of Li2NH produced Li, N2 and H2 via two steps: Li2NH into an intermediate species --- Li4NH and then into Li. The kinetic analysis of Li2NH decomposition showed that the activation energies are 533.6 kJ/mol for the first step and 754.2 kJ/mol for the second step. Furthermore, XRD demonstrated that the Li4NH, which was generated in the decomposition of Li2NH, formed a solid solution with Li2NH. In the solid solution, Li4NH possesses a similar cubic structure as Li2NH. The lattice parameter of the cubic Li4NH is 0.5033nm. The decompositions of LiNH2 and Li2NH can be promoted by chloride ion (Cl-). The introduction of Cl- into LiNH2 resulted in the generation of a new NH3 peak at low temperature of 250 °C besides the original NH3 peak at 330 °C in TPD profiles. Furthermore, Cl- can decrease the decomposition temperature of Li2NH by about 110 °C. The degradation of Li3N was systematically investigated with techniques of XRD, Fourier transform infrared (FT-IR) spectroscopy, and UV-visible spectroscopy. It was found that O2 could not affect Li3N at room temperature. However, H2O in air can cause the degradation of Li3N due to the reaction between H2O and Li3N to LiOH. The produced LiOH can further react with CO2 in air to Li2CO3 at room temperature. Furthermore, it was revealed that Alfa-Li3N is more stable in air than Beta-Li3N. The chemical stability of Li13N4Br in air has been investigated by XRD, TPD-MS, and UV-vis absorption as a function of time. The aging process finally leads to the degradation of the Li13N4Br into Li2CO3, lithium bromite (LiBrO2) and the release of gaseous NH3. The reaction order n = 2.43 is the best fitting for the Li13N4Br degradation in air reaction. Li13N4Br energy gap was calculated to be 2.61 eV.