Since 2008, more than half of the world population lives in cities. To cope with this rapid urbanization in a sustainable manner, transit systems all around the world are likely to grow. By studying 33 networks in the world, this thesis identifies the properties and effects of metro network designs by using a graph theory approach. After the literature review, a new methodology was introduced to translate networks into graphs; it notably accounts for various transit specificities (e.g., presence of lines). Metro networks were then characterised according to their State, Form, and Structure; where State relates to the development phase of metros; Form investigates the link between metros and the built environment; Structure examines the intrinsic properties of metros, by notably looking at their connectivity. Subsequently, the complexity and robustness of metros were studied; metros were found to possess scale-free and small-world features although showing atypical topologies; robustness emphasizes on the presence of alternative paths. Three network design indicators (coverage, directness and connectivity) were then related to ridership (annual boardings per capita), and positive relations were observed, which suggests that network design plays an important role in their success. Finally, these concepts were applied to the Toronto metro plans announced by the Toronto regional transportation authority, Metrolinx; it was found that the grid-pattern nature of the plans could hinder the success of the metro; seven possible improvements were suggested. Overall, the topology of metro networks can play a key role in their success. The concepts presented here can particularly be useful to transit planners; they should also be used along with conventional planning techniques. New transit projects could benefit greatly from an analysis of their network designs, which in turn may play a relevant role in the global endeavour for sustainability.

Datacenters are a significant capital expense for many enterprises. Yet, they are difficult to manage and are hard to design and maintain. The initial design of a datacenter network tends to follow vendor guidelines, but subsequent upgrades and expansions to it are mostly ad hoc, with equipment being upgraded piecemeal after its amortization period runs out and equipment acquisition is tied to budget cycles rather than changes in workload. These networks are also brittle and inflexible. They tend to be manually managed, and cannot perform dynamic traffic engineering. The high-level goal of this dissertation is to reduce the total cost of owning a datacenter by improving its network. To achieve this, we make the following contributions. First, we develop an automated, theoretically well-founded approach to planning cost-effective datacenter upgrades and expansions. Second, we propose a scalable traffic management framework for datacenter networks. Together, we show that these contributions can significantly reduce the cost of operating a datacenter network. To design cost-effective network topologies, especially as the network expands over time, updated equipment must coexist with legacy equipment, which makes the network heterogeneous. However, heterogeneous high-performance network designs are not well understood. Our first step, therefore, is to develop the theory of heterogeneous Clos topologies. Using our theory, we propose an optimization framework, called LEGUP, which designs a heterogeneous Clos network to implement in a new or legacy datacenter. Although effective, LEGUP imposes a certain amount of structure on the network. To deal with situations when this is infeasible, our second contribution is a framework, called REWIRE, which using optimization to design unstructured DCN topologies. Our results indicate that these unstructured topologies have up to 100-500\% more bisection bandwidth than a fat-tree for the same dollar cost. Our third contribution is two frameworks for datacenter network traffic engineering. Because of the multiplicity of end-to-end paths in DCN fabrics, such as Clos networks and the topologies designed by REWIRE, careful traffic engineering is needed to maximize throughput. This requires timely detection of elephant flows---flows that carry large amount of data---and management of those flows. Previously proposed approaches incur high monitoring overheads, consume significant switch resources, or have long detection times. We make two proposals for elephant flow detection. First, in the Mahout framework, we suggest that such flows be detected by observing the end hosts' socket buffers, which provide efficient visibility of flow behavior. Second, in the DevoFlow framework, we add efficient stats-collection mechanisms to network switches. Using simulations and experiments, we show that these frameworks reduce traffic engineering overheads by at least an order of magnitude while still providing near-optimal performance.

A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines the movement of our particle. Quantum walks have found a number of algorithmic applications, including unstructured search, element distinctness and Boolean formula evaluation. We will examine the properties of periodicity and state transfer. In particular, we will prove a result of the author along with Godsil, Kirkland and Severini, which states that pretty good state transfer occurs in a path of length n if and only if the n+1 is a power of two, a prime, or twice a prime. We will then examine the property of strong cospectrality, a necessary condition for pretty good state transfer from u to v. We will then consider quantum walks involving more than one particle. In addition to moving around the graph, these particles interact when they encounter one another. Varying the nature of the interaction term gives rise to a range of different behaviours. We will introduce two graph invariants, one using a continuous-time multi-particle quantum walk, and the other using a discrete-time quantum walk. Using cellular algebras, we will prove several results which characterize the strength of these two graph invariants. Let A be an association scheme of n × n matrices. Then, any element of A can act on the space of n × n matrices by left multiplication, right multiplication, and Schur multiplication. The set containing these three linear mappings for all elements of A generates an algebra. This is an example of a Jaeger algebra. Although these algebras were initially developed by Francois Jaeger in the context of spin models and knot invariants, they prove to be useful in describing multi-particle walks as well. We will focus on triply-regular association schemes, proving several new results regarding the representation of their Jaeger algebras. As an example, we present the simple modules of a Jaeger algebra for the 4-cube.

This thesis treats some of the problems related to the good drawings D$ sb{ rm n}$ of the complete graph K$ sb{ rm n}$. The first of these problems is obtaining all the non-isomorphic good drawings D$ sb{ rm n}$ of K$ sb{ rm n}$. After conjecturing that any good drawing D$ sb{ rm n}$ of K$ sb{ rm n}$ has at least one crossing-free Hamiltonian Circuit, an algorithm generating all the non-isomorphic good drawings D$ sb{ rm n}$ of K$ sb{ rm n}$ is developed. The second problem, determining the existence of a rectilinear drawing D$ sb{ rm n}$ of K$ sb{ rm n}$ with a given set of crossings, is solved by finding a characteristic of the rectilinear drawings D$ sb{ rm n}$ of K$ sb{ rm n}$. An algorithm using this characteristic determines whether a given set of crossing defines a rectilinear drawing D$ sb{ rm n}$ of K$ sb{ rm n}$. The last problem, to generate all the non-isomorphic rectilinear drawings D$ sb{ rm n}$ of K$ sb{ rm n}$, is solved by an algorithm using a set of rectilinear drawings D$ sb{ rm n-1}$ of K$ sb{ rm n-1}$.

Image segmentation has an important role in image processing because it is a tool to obtain higher level object descriptions for further processing. In some applications such as large image databases or video image sequence segmentations, the speed of the segmentation algorithm may become a drawback of the application. This thesis work is a study to improve the run-time performance of a well-known segmentation algorithm, namely the Recursive Shortest Spanning Tree (RSST). Both the original and the fast RSST found in the literature are analyzed and a comparison is made between these techniques. Simple modifications and an alternative link cost structure are proposed and evaluated. Finally, a distributed implementation based on a simple image partitioning strategy is attempted. The thesis presents the results of an extensive computational study with respect to both run-time performance and image segmentation quality.

This thesis contains several results concerning the combinatorics of partially ordered sets (posets) which are either of enumerative or extremal nature. <br><br> The first concerns conjectures of Friedland and Kahn, which state that the (extremal) d-regular graph on N vertices containing both the maximal number of matchings and independent sets of a fixed size is the graph consisting of disjoint union of appropriate number of complete bipartite d-regular graphs on 2d vertices. We show that the conjectures are true in an asymptotic sense, using entropy techniques. <br><br> As a second result, we give tight bounds on the size of the largest Boolean family which contains no three distinct subsets forming an "induced V" (i.e. if A,B,C are all in our family, if C is contained in the intersection of A B, A must be a subset of B). This result, though similar to known results, gives the first bound on a family defined by an induced property. <br><br> We pose both Dedekind-type questions concerning the number of antichains and a Stanley-type question concerning the number of linear extensions in generalized Boolean lattices; namely, products of chain posets and the poset of partially defined functions. We provide asymptotically tight bounds for these problems. <br><br> A Boolean function, f, is called cherry-free if for all triples x,y,z where z covers both x and y, f(z)=1 whenever both f(x)=1 and f(y)=1. We give bounds on the number of cherry-free functions on bipartite regular posets, with stronger results for bipartite posets under an additional co-degree hypotheses. We discuss applications of these functions to Boolean Horn functions and similar structures in ranked regular posets.

The objective of this research is to explore high-speed analog-to-digital converters (ADCs) using silicon-germanium (SiGe) heterojunction bipolar transistors (HBTs) for wireless digital receiver applications. The stringent requirements of ADCs for the high-performance next-generation wireless digital receiver include (1) low power, (2) low cost, (3) wide input signal bandwidth, (4) high sampling rate, and (5) medium to high resolution. The proposed research achieves the objective by implementing high-performance ADC's key building blocks and integrating these building blocks into a complete sigma-delta analog-to-digital modulator that satisfies the demanding specifications of next-generation wireless digital receiver applications. The scope of this research is divided into two main parts: (1) high-performance key building blocks of the ADC, and (2) high-speed sigma-delta analog-to-digital modulator. The research on ADC's building blocks includes the design of two high-speed track-and-hold amplifiers (THA) and two wide-bandwidth comparators operating at the sampling rate > 10 GS/sec with satisfying resolution. The research on high-speed sigma-delta analog-to-digital modulator includes the design and experimental characterization of a high-speed second-order low-pass sigma-delta modulator, which can operate with a sampling rate up to 20 GS/sec and with a medium resolution. The research is envisioned to demonstrate that the SiGe HBT technology is an ideal platform for the design of high-speed ADCs.

The objective of this dissertation is to study commonly occurring location and clustering problems on graphs. The dissertation is presented as a collection of results in topics including finding maximum cliques in large graphs, graph clustering in large scale graphs, determining location of facilities for pre-positioning emergency relief supplies, and selecting nodes to form a virtual backbone in a wireless sensor network. To begin with, a new clique relaxation called a k-community is defined as a connected subgraph such that endpoints of every edge have at least k common neighbors within the subgraph. It is used to develop scale reduction techniques to obtain the maximum clique on very large scale real life networks. Analytically, the technique is been shown to be very effective on power-law random graphs. Experimental results on real life graph instances (Collaboration networks, P2P networks, Social networks, etc.) show our procedure to be much more effective than a regular k-core peeling approach. Next, a general purpose network clustering algorithm based on the clique relaxation concept of k-community is presented. A salient feature of this approach is that it does not use any prior information about the structure of the network. By defining a cluster as a k-community, the proposed algorithm aims to provide a clustering of a network into k-communities with varying values of k. Even though the algorithm is not designed to optimize any particular performance measure, the computational results suggest that it performs well on a number of criteria that are used in literature to evaluate the quality of a clustering. The third topic deals with choosing the locations of disaster response facilities for the storage of emergency supplies, which is critical to the quality of service provided in a large scale emergency like an earthquake. In the existing literature, large scale emergency facility location models have either assumed that disaster response facilities will always be functioning and available when required, or that the functioning of a facility is independent of a particular disaster scenario. In this paper new location models are presented that explicitly take into consideration the stochastic nature of the impact a disaster can have on the disaster response facilities and the population centers in surrounding areas. A comparison of the results obtained using our models with those from models available in literature using a case study suggests that the locations suggested by the model in this paper significantly reduce the expected cost of transportation of supplies when we consider the damage a disaster causes to the disaster response facilities and areas near it. Lastly, a distributed approximate algorithm for forming the communication backbone in wireless sensor networks is presented. Some of the most popular routing protocols for wireless sensor networks require a virtual backbone for efficient communication be- tween the sensors. Connected Dominating Sets (CDS) have been studied as a method of choosing nodes to be in the backbone. The traditional approach is to assume that the transmission range of each node is given and then minimize the number of nodes in the CDS representing the backbone. A recently introduced alternative strategy is based on the concept of k-bottleneck connected dominating set (k-BCDS), which, given a positive integer k, minimizes the transmission range of the nodes that ensures a CDS of size k exists in the network. This paper provides a 6-approximate distributed algorithm for the k-BCDS problem. The results of empirical evaluation of the proposed algorithm are also included.

"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling." / Doctor of Philosophy

The research presented in this thesis investigates the performance of some well-known heuristic algorithms on graph decomposition problems. First, a genetic algorithm is introduced and some modifications are trialled on finding Steiner triple systems (STS) of small orders. The results show that traditional genetic algorithms are not well suited to finding graph decompositions. Then a hill climbing optimisation technique is presented and investigated in the context of cycle decompositions. Such searches have previously proved to be effective at finding STSs. However, the general hill climbing approach is not immediately applicable to decompositions into cycles of length larger than 3. A modification of the hill climbing algorithm for cycles, called slippery hill climbing, is introduced and tested on decompositions of graphs into cycles of small lengths larger than 3. Slippery hill climbing successfully decomposed complete and dense non-complete graphs of considerable sizes into cycles of small lengths. In addition, we applied the slippery hill climbing approach to completing partial latin squares. It is reasonably expected that the algorithms developed in this study will also be applicable to other related problems in combinatorics and graph theory.