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.

High temperature, low cycle fatigue of a hybrid particulate/fiber aluminum metal-matrix composite

The effect of shot particles on the high temperature, low cycle fatigue of a hybrid fiber/particulate metal-matrix composite (MMC) was studied. Two hybrid composites with the general composition A356/35%SiC particle/5%Fiber (one without shot) were tested. It was found that shot particles acting as stress concentrators had little effect on the fatigue performance. It appears that fibers with a high silica content were more likely to debond from the matrix. Final failure of the composite was found to occur preferentially in the matrix. SiC particles fracture progressively during fatigue testing, leading to higher stress in the matrix, and final failure by matrix overload. A continuum mechanics based model was developed to predict failure in fatigue based on the tensile properties of the matrix and particles. By accounting for matrix yielding and recovery, composite creep and particle strength distribution, failure of the composite was predicted.

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.

Extracellular matrix of the charophycean green algae

A comprehensive knowledge of cell wallstructure and function throughout the plant kingdom is essential to understanding cell wall evolution. The fundamental understanding of the charophycean green algal cell wall is broadening. The similarities and differences that exist between land plant and algal cell walls provide opportunities to understand plant evolution. A variety of polymers previously associated with higher plants were discovered in the charophycean green algae (CGA), including homogalacturonans, cross-linking glycans, arabinogalactan protein, β-glucans, and cellulose. The cellulose content of CGA cell walls ranged from 6% to 43%, with the higher valuescomparable to that found in the primary cell wall of land plants (20-30%). (1,3)β-glucans were found in the unicellular Chlorokybus atmophyticus, Penium margaritaceum, and Cosmarium turpini, the unbranched filamentous Klebsormidium flaccidum, and the multicellular Chara corallina. The discovery of homogalacturonan in Penium margaritaceum representsthe first confirmation of land plant-type pectinsin desmids and the second rigorous characterization of a pectin polymer from the charophycean algae. Homogalacturonan was also indicated from the basal species Chlorokybus atmophyticus and Klebsormidium flaccidum. There is evidence of branched pectins in Cosmarium turpini and linkage analysis suggests the presence of type I rhamnogalacturonan (RGI). Cross-linking β-glucans are associated with cellulose microfibrils during land plant cell growth, and were found in the cell wall of CGA. The evidence of mixed-linkage glucan (MLG) in the 11 charophytesis both suprising and significant given that MLG was once thought to be specific to some grasses. The organization and structure of Cosmarium turpini and Chara corallina MLG was found to be similar to that of Equisetumspp., whereas the basal species of the CGA, Chlorokybus atmophyticus and Klebsormidium flaccidum, have unique organization of alternating of 3- and 4-linkages. The significance of this result on the evolution of the MLG synthetic pathway has yet to be determined. The extracellular matrix (ECM) of Chlorokybus atmophyticus, Klebsormidium flaccidum, and Spirogyra spp. exhibits significant biochemical diversity, ranging from distinct “land plant” polymers to polysaccharides unique to these algae. The neutral sugar composition of Chlorokybus atmophyticus hot water extract and Spirogyra extracellular polymeric substance (EPS), combined with antibody labeling results, revealed the distinct possibility of an arabinogalactan protein in these organisms. Polysaccharide analysis of Zygnematales (desmid) EPS, indicated a probable range of different EPS backbones and substitution patterns upon the core portions of the molecules. Desmid EPS is predominately composed of a complex matrix of branched, uronic acid containing polysaccharides with ester sulfate substitutions and, as such, has an almost infinite capacity for various hydrogen bonding, hydrophobic interaction and ionic cross-bridging motifs, which characterize their unique function in biofilms. My observations support the hypothesis that members of the CGA represent the phylogenetic line that gave rise to vascular plants and that the primary cell wall of vascular plants many have evolved directly from structures typical of the cell wall of filamentous green algae found in the charophycean green algae.