The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides a characterization of planar graphs in terms of their poset dimension, as follows: a graph G is planar if and only if the dimension of the incidence poset of G is at most three. One of the implications of the theorem is proved by giving an explicit mapping of the vertices to R^2 that defines a straightline embedding of the graph. The other implication is proved by introducing the concept of normal labelling. Normal labellings of plane triangulations can be used to obtain a realizer of the incidence poset. We present an exposition of Schnyder’s theorem with his original proof, using normal labellings. An alternate proof of Schnyder’s Theorem is also presented. This alternate proof does not use normal labellings, instead we use some structural properties of a realizer of the incidence poset to deduce the result. Some applications and a generalization of one implication of Schnyder’s Theorem are also presented in this work. Normal labellings of plane triangulations can be used to obtain a barycentric embedding of a plane triangulation, and they also induce a partition of the edge set of a plane triangulation into edge disjoint trees. These two applications of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and canonical orderings to obtain a compact drawing of a graph, are also presented. A generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s Theorem was proved by Ossona de Mendez. This generalization is also presented in this work. The concept of order labelling is also introduced and we show some similarities of the order labelling and the normal labelling. Finally, we conclude this work by showing the source code of some implementations done in Sage.

It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.

For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence of a G. An equivalent definition exists for non-orientable surfaces. In 1983, Jozef Širáň proved that for every decreasing, convex sequence of non-negative integers, there is a graph G such that this sequence is the crossing sequence of G. This main result of this thesis proves the existence of a graph with non-convex crossing sequence of arbitrary length.

Complex networks like the Internet, peer-to-peer systems, and emerging sensor and ad-hoc networks are large distributed decentralized communication systems arising repeatedly in today's technology. In such networks it is critical to characterize network performance as the size of the network scales. The focus of my work is to relate basic network performance metrics to structural characteristics of underlying network topologies, and to develop protocols that reinforce and exploit desired structural characteristics. For the case of the Internet at the Autonomous System level, we relate the graph theoretic notions of conductance and spectrum to network clustering and network congestion. In particular, we show how spectral analysis can identify clusters, and how the presence of clusters affects congestion. This is important for network prediction and network simulation. For the case of peer-to-peer networks we relate conductance and spectral gap to the fundamental questions of searching and topology maintenance. We propose new protocols for maintaining peer-to-peer networks with good conductance and low network overhead. We compare the performance of the traditional method of search by flooding to searching by random walks. We isolate cases of practical interest, such as clustered and dynamic network topologies, where the latter have superior performance. The improvement in the performance can be directly quantified in terms of the conductance of the underlying topology. We introduce further hybrid search schemes, of which flooding and random walks are special instances, which aim to improve the performance of searching by using locally maintained information about the network topology.

A graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In this thesis, we study the structure of critical graphs on higher surfaces. One major result in this area is Carsten Thomassen's proof that there are finitely many 6-critical graphs on a fixed surface. This proof involves a structural theorem about a precolored cycle C of length q. In general terms, he proves that a coloring, c, of C, can be extended inside the cycle, or there exists a subgraph H with at most a number of vertices exponential in q such that c can not be extended to a 5-coloring of H. In Chapter 2, we proved an alternative proof that reduces the number of vertices in H to be cubic in q. In Chapter 3, we find the nine 6-critical graphs among all graphs embeddable on the Klein bottle. In Chapter 4, we prove a result concerning critical graphs related to an analogue of Steinberg's conjecture for higher surfaces. We show that if G is a 4-critical graph embedded on surface S, with Euler genus g and has no cycles of length four through ten, then G has at most 2442g + 37 vertices.

Classical partitioning problems, like the Max-Cut problem, ask for partitions that optimize one quantity, which are important to such fields as VLSI design, combinatorial optimization, and computer science. Judicious partitioning problems on graphs or hypergraphs ask for partitions that optimize several quantities simultaneously. In this dissertation, we work on judicious partitions of graphs and hypergraphs, and solve or asymptotically solve several open problems of Bollobas and Scott on judicious partitions, using the probabilistic method and extremal techniques.

Thesis (M. Sc.)--University of Canterbury, 2009. / Typescript (photocopy). Includes bibliographical references (leaves 73-77) and index. Also available via the World Wide Web.

Thèse doctorat : Informatique : Villeurbanne, INSA : 2005. Thèse doctorat : Informatique : Universidad de las Américas - Puebla : 2005. / Thèse soutenue en co-tutelle. Thèse rédigée en français, en anglais et en espagnol. Titre provenant de l'écran-titre. Bibliogr. p. 174-182.

Thesis (M.Sc.)--Memorial University of Newfoundland, 2001. / Includes index. Bibliography: leaf 41.

Thesis (M.S.)--University of Missouri-Columbia, 2003. / Typescript. Includes bibliographical references (leaves 70-74). Also available on the Internet.