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.

ANALYSIS AND VISUALIZATION OF FLOW FIELDS USING INFORMATION-THEORETIC TECHNIQUES AND GRAPH-BASED REPRESENTATIONS

Three-dimensional flow visualization plays an essential role in many areas of science and engineering, such as aero- and hydro-dynamical systems which dominate various physical and natural phenomena. For popular methods such as the streamline visualization to be effective, they should capture the underlying flow features while facilitating user observation and understanding of the flow field in a clear manner. My research mainly focuses on the analysis and visualization of flow fields using various techniques, e.g. information-theoretic techniques and graph-based representations. Since the streamline visualization is a popular technique in flow field visualization, how to select good streamlines to capture flow patterns and how to pick good viewpoints to observe flow fields become critical. We treat streamline selection and viewpoint selection as symmetric problems and solve them simultaneously using the dual information channel [81]. To the best of my knowledge, this is the first attempt in flow visualization to combine these two selection problems in a unified approach. This work selects streamline in a view-independent manner and the selected streamlines will not change for all viewpoints. My another work [56] uses an information-theoretic approach to evaluate the importance of each streamline under various sample viewpoints and presents a solution for view-dependent streamline selection that guarantees coherent streamline update when the view changes gradually. When projecting 3D streamlines to 2D images for viewing, occlusion and clutter become inevitable. To address this challenge, we design FlowGraph [57, 58], a novel compound graph representation that organizes field line clusters and spatiotemporal regions hierarchically for occlusion-free and controllable visual exploration. We enable observation and exploration of the relationships among field line clusters, spatiotemporal regions and their interconnection in the transformed space. Most viewpoint selection methods only consider the external viewpoints outside of the flow field. This will not convey a clear observation when the flow field is clutter on the boundary side. Therefore, we propose a new way to explore flow fields by selecting several internal viewpoints around the flow features inside of the flow field and then generating a B-Spline curve path traversing these viewpoints to provide users with closeup views of the flow field for detailed observation of hidden or occluded internal flow features [54]. This work is also extended to deal with unsteady flow fields. Besides flow field visualization, some other topics relevant to visualization also attract my attention. In iGraph [31], we leverage a distributed system along with a tiled display wall to provide users with high-resolution visual analytics of big image and text collections in real time. Developing pedagogical visualization tools forms my other research focus. Since most cryptography algorithms use sophisticated mathematics, it is difficult for beginners to understand both what the algorithm does and how the algorithm does that. Therefore, we develop a set of visualization tools to provide users with an intuitive way to learn and understand these algorithms.

DESIGN AND SYNTHESIS OF MULTIFUNCTIONAL POLYMERIC MICELLES FOR GENE-DIRECTED ENZYME PRODRUG THERAPY

Gene-directed enzyme prodrug therapy is a form of cancer therapy in which delivery of a gene that encodes an enzyme is able to convert a prodrug, a pharmacologically inactive molecule, into a potent cytotoxin. Currently delivery of gene and prodrug is a two-step process. Here, we propose a one-step method using polymer nanocarriers to deliver prodrug, gene and cytotoxic drug simultaneously to malignant cells. Prodrugs acyclovir, ganciclovir and 5-doxifluridine were used to directly to initiate ring-opening polymerization of epsilon-caprolactone, forming a hydrophobic prodrug-tagged poly(epsilon-caprolactone) which was further grafted with hydrophilic polymers (methoxy poly(ethylene glycol), chitosan or polyethylenemine) to form amphiphilic copolymers for micelle formation. Successful synthesis of copolymers and micelle formation was confirmed by standard analytical means. Conversion of prodrugs to their cytotoxic forms was analyzed by both two-step and one-step means i.e. by first delivering gene plasmid into cell line HT29 and then challenging the cells with the prodrug-tagged micelle carriers and secondly by complexing gene plasmid onto micelle nanocarriers and delivery gene and prodrug simultaneously to parental HT29 cells. Anticancer effectiveness of prodrug-tagged micelles was further enhanced by encapsulating chemotherapy drugs doxorubicin or SN-38. Viability of colon cancer cell line HT29 was significantly reduced. Furthermore, in an effort to develop a stealth and targeted carrier, CD47-streptavidin fusion protein was attached onto the micelle surface utilizing biotin-streptavidin affinity. CD47, a marker of self on the red blood cell surface, was used for its antiphagocytic efficacy, results showed that micelles bound with CD47 showed antiphagocytic efficacy when exposed to J774A.1 macrophages. Since CD47 is not only an antiphagocytic ligand but also an integrin associated protein, it was used to target integrin alpha(v)beta(3), which is overexpressed on tumor-activated neovascular endothelial cells. Results showed that CD47-tagged micelles had enhanced uptake when treated to PC3 cells which have high expression of alpha(v)beta(3). The synthesized multifunctional polymeric micelle carriers developed could offer a new platform for an innovative cancer therapy regime.