Ph.D. (Mathematics) / Please refer to full text to view abstract

The n-antiprism graph is not widely studied as a class of graphs in graph theory hence there is not much literature. We begin by de ning the n-antiprism graph and discussing properties, which we prove in the thesis, and which have not been previously presented in graph theory literature. Some of our signi cant results include proving that an n-antiprism is 4-connected, 4-edge connected and has a pathwidth of 4. A highly studied area of graph theory is the chromatic polynomial of graphs. We investigate the chromatic polynomial of the antiprism graph and attempt to nd explicit expressions for the chromatic polynomial of the antiprism graph. We express this chromatic polynomial in several forms to discover the best-suited form. We then explore the Tutte polynomial and search for an explicit expression of the Tutte polynomial of the antiprism graph. Using the relationship between a graph and its dual graph, we provide an iterative expression of the Tutte polynomial of the antiprism graph.

Lau, Siu Cheong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 51-52). / Abstracts in English and Chinese. / Chapter 1 --- In search of minimal cycles --- p.9 / Chapter 1.1 --- What do we mean by cycles? --- p.9 / Chapter 1.2 --- Integral currents --- p.10 / Chapter 1.3 --- Calibration theory --- p.13 / Chapter 2 --- Motivation from the Hodge Conjecture --- p.17 / Chapter 2.1 --- Hodge theory on Riemannian manifolds --- p.17 / Chapter 2.2 --- Hodge decomposition in Kahler manifolds --- p.19 / Chapter 2.3 --- The Hodge conjecture --- p.22 / Chapter 3 --- Variation of cycles in symmetric orbit --- p.26 / Chapter 3.1 --- Variational formulae --- p.26 / Chapter 3.2 --- Stability of cycles in Sm and CPn --- p.29 / Chapter 3.3 --- Symmetric orbit in Euclidean space --- p.31 / Chapter 3.4 --- Projective spaces in simple Jordan algebra --- p.39 / Chapter 3.4.1 --- Introduction to simple Jordan algebra --- p.39 / Chapter 3.4.2 --- Projective spaces as symmetric orbits --- p.41 / Chapter 3.4.3 --- Computation of second fundamental form --- p.43 / Chapter 3.4.4 --- The main theorem --- p.45 / Chapter 3.5 --- Future directions --- p.49 / Bibliography --- p.51

No description available.

No description available.

No description available.

Mathematical concepts have aided the progression of many different fields of study. Math is not only helpful in science and engineering, but also in the humanities and social sciences. Therefore, it seemed quite natural to apply my preliminary work with set intersections to voting theory, and that application has helped to focus my thesis. Rather than studying set intersections in general, I am attempting to study set intersections and what they mean in a voting situation. This can lead to better ways to model preferences and to predict which campaign platforms will be most popular. Because I feel that allowing people to only vote for one candidate results in a loss of too much information, I consider approval voting, where people can vote for as many platforms as they like.

Kolmogorov complexity is a theory based on the premise that the complexity of a binary string can be measured by its compressibility; that is, a string’s complexity is the length of the shortest program that produces that string. We explore applications of this measure to graph theory.

Relational information visualisation concerns viewing relational data, where the underlying data model is a graph. Hierarchical visualisation is one of hot topics in graph visualisation in which the data is organised in a hierarchical structure. As the amount of information, that we want to visualise, becomes larger and the relations become more complex, classical visualisation techniques and hierarchical drawing methods tend to be inadequate.Traditional hierarchical visualisation algorithms are more concerned with the readability of the layouts. They usually do not consider the efficient utilisation of the geometrical plane for the drawings. Therefore, for most hierarchical layouts, a large portion of display space is wasted as background. The aim of this research is to investigate a space-efficient approach to handle the visualisation of large hierarchies in two-dimensional spaces. This thesis introduces a new graph visualisation approach called enclosure+ connection for visualizing large hierarchies. This approach maximises the space utilisation by taking advantages of the traditional enclosure partitioning approach, while it retains the display of a traditional node-link diagram to hopefully provide users a direct perception of relational structures. The main contribution of this thesis is layout and navigation algorithms for visualising large hierarchies. Two layout algorithms, the space-optimised tree and the EncCon tree, have been developed to achieve the space-efficient visualisation. Both algorithms use the enclosure concept to define layout of hierarchies, which ensure the efficient utilisation of display space. Two focus+context navigation and interaction methods have been proposed to cooperate with the visualization of large hierarchies. Several advanced computer graphics approaches, such as graphic distortion and transparency, are used for the development of these navigation methods. Two case studies have been implemented to evaluate the layout algorithms and the associated navigation methods. The first case study is an application of a shared collaborative workspace which aims to provide users with a better assistance for visual manipulation and navigation of knowledge-based information. The second case study is a visual browser for navigating large-scale online product catalogues. Although the case studies have provided some useful evaluation, formal usability studies would be required to justify fully the effectiveness of these layout and navigation methods. Although this task has not carried out in this research, the author has presented his usability study's plan as a future work.

Thesis (Ph. D.)--Lehigh University, 2000. / Includes vita. Includes bibliographical references (leaves 142-147).