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.

Uptake and fate of hexahydro-1,3,5-trinitro-1,3,5-triazine by Chrysopogon zizanioides

Hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) is a nitramine compound that has been used heavily by the military as an explosive. Manufacturing, use, and disposal of RDX have led to several contamination sites across the United States. RDX is both persistent in the environment and a threat to human health, making its remediation vital. The use of plants to extract RDX from the soil and metabolize it once it is in the plant tissue, is being considered as a possible solution. In the present study, the tropical grass Chrysopogon zizanioides was grown hydroponically in the presence RDX at 3 different concentration levels: 0.3, 1.1, and 2.26 ppm. The uptake of RDX was quantified by high performance liquid chromatography (HPLC) analysis of media samples taken every 6 hr during the first 24 hr and then daily over a 30-day experimental period. A rapid decrease in RDX concentration in the media of both controls and plant treatments was seen within the first 18 hours of the experiment with the greatest loss in RDX over time occurring within the first 6 hours of exposure. The loss was similar in both controls and plant exposures and possibly attributed to rapid uptake by the containers. A plant from one treatment at each of the three concentrations was harvested at Day 10, 20 and 30 throughout the experiment and extracted to determine the localization of RDX within the tissue and potentially identify any metabolites on the basis of differing retention times. Of the treatments containing 0.3, 1.1, and 2.26 ppm RDX, 13.1%, 18.3%, and 24.2% respectively, was quantified in vetiver extracts, with the majority of the RDX being localized to the roots. All plants not yet harvested were harvested on Day 30 of the experiment. A total of three plants exposed to each concentration level as well as the control, were extracted and analyzed with HPLC to determine amount of RDX taken up, localization of RDX within the plant tissue, and potentially identify any metabolites. Phytotoxicity of RDX to vetiver was also monitored. While a loss in biomass was observed in plants exposed to all the different concentrations of RDX, control plants grown in media not exposed to RDX showed the greatest biomass loss of all the treatments. There was also little variation in chlorophyll content between the different concentration treatments with RDX. This preliminary greenhouse study of RDX uptake 10 by Chrysopogon zizanioides will help indicate the potential ability of vetiver to serve as a plant system in the phytoremediation of RDX.

Syntheses and characterization of monomeric Mo(VI) complexes with bidentate phosphine oxide ligands and dimeric and tetrameric Mo(V) clusters with benzoic acid and phosphinic acid derivatives, containing MoO2, Mo2O2(μ-O)2 and Mo4O4(μ3-O)4

Mo(VI) oxo complexes have been persistently sought after as epoxidation catalysts. Further, Mo(V) oxo clusters of the form M4(µ3-X)4 (M = transition metal, X = O, S) have been rigorously studied due to their remarkable structures and also their usefulness as models for electronic studies. The syntheses and characterizations of new Mo(VI) and Mo(V) oxo complexes have been described in this dissertation. Two new complexes MoO2Cl2Ph2P(O)CH2COOH and MoO2Cl2Ph2P(O)C6H4tBuS(O) were synthesized from reactions of “MoO2Cl2” with ligands Ph2P(O)CH2COOH and Ph2P(O)C6H4tBuS(O). Tetrameric packing arrangements comprised of hydrogen bonds were obtained for the complex MoO2Cl2Ph2P(O)CH2COOH and the ligand Ph2P(O)CH2COOH. Further the stability of an Mo-O bond was preferred over the Mo-S bond even though this resulted in the formation of a more strained seven membered ring. Tetranuclear Mo(V) complexes of the form [Mo4(µ3-O)4(µ-O2PR2)4O4], (PR2 = PPh2, PMe2) were synthesized using reactions of MoO2(acac)2 with diphenyl and dimethyl phosphinic acids, in ethanol. In the crystal structure of these complexes four Mo=O units are interconnected by four triply bridging oxygen atoms and bridging phosphinate ligands. The complex exhibited fourfold symmetry as evidenced by a single 31P NMR peak for the P atoms in the coordinated ligands. Reaction of WO2(acac)2 with Ph2POOH in methanol resulted in a dimeric W(VI) complex [(CH3O)2(O)W(µ-O)( µ-O2PPh2)2W(O)(CH3O)2] which contained a packing disorder in its crystal structure. Similar reactions of MoO2(acac)2 with benzoic acid derivatives resulted in dimeric complexes of the form [Mo2O2(acac)2(µ-O)(µ-OC2H5)(µ-O2CR)] (R = C6H5, (o-OH)C6H4, (p-Cl)C6H4, (2,4-(OH)2)C6H3, (o-I)C6H4) and one tetrameric complex [Mo2O2(acac)2(µ-O)(µ-OC2H5)(µ-O2C)C6H4(p-µ-O2C)Mo2O2(acac)2(µ-O)(µ-OC2H5)] with terephthalic acid. 1H NMR proved very useful in the prediction of the formation of dimers with the substituted benzoic acids, which were also confirmed by elemental analyses. The reductive capability of ethanol proved instrumental in the syntheses of Mo(V) tetrameric and dimeric clusters. Synthetic details, IR, 1H and 31P NMR spectroscopy and elemental analyses are reported for all new complexes. Further, single crystal X-ray structures of MoO2Cl2Ph2P(O)CH2COOH, MoO2Cl2Ph2P(O)C6H4tBuS(O), [Mo4(µ3-O)4(µ-O2PR2)4O4], (PR2 = PPh2, PMe2), [(CH3O)2(O)W(µ-O)( µ-O2PPh2)2W(O)(CH3O)2] and [Mo2O2(acac)2(µ-O)(µ-OC2H5)(µ-O2CR)] (R = C6H5, (o-OH)C6H4) are also presented.