Three Existence Problems in Extremal Graph Theory

Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics. In this thesis we take this approach to three different structural questions rooted in extremal graph theory. When studying graph representations, we seek efficient ways to encode the structure of a graph. For example, an {\it interval representation} of a graph $G$ is an assignment of intervals on the real line to the vertices of $G$ such that two vertices are adjacent if and only if their intervals intersect. We consider graphs that have {\it bar $k$-visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations. We obtain results on $\mathcal{F}_k$, the family of graphs having bar $k$-visibility representations. We also study $\bigcup_{k=0}^{\infty} \mathcal{F}_k$. In particular, we determine the largest complete graph having a bar $k$-visibility representation, and we show that there are graphs that do not have bar $k$-visibility representations for any $k$. Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs. Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around $G$ provided that each move transfers weight from a vertex to a heavier neighbor. Our goal in making acquisition moves is to consolidate all of the weight in $G$ on the minimum number of vertices; this minimum number is the {\it acquisition number} of $G$. We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight. We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter $2$. We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex. Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects. Some local conditions are so limiting that very few objects satisfy the requirements. For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor. Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex. We study a related structural restriction where similar phenomena occur. For a fixed graph $H$, we consider those graphs that do not contain $H$ and such that the addition of any edge completes exactly one copy of $H$. Such a graph is called {\it uniquely $H$-saturated}. We study the existence of uniquely $H$-saturated graphs when $H$ is a path or a cycle. In particular, we determine all of the uniquely $C_4$-saturated graphs; there are exactly ten. Interestingly, the uniquely $C_{5}$-saturated graphs are precisely the friendship graphs characterized by Wilf

Phasor Measurement Units Optimal Placement and Performance Limits for Fault Localization

In this paper, the performance limits of faults localization are investigated using synchrophasor data. The focus is on a non-trivial operating regime where the number of Phasor Measurement Unit (PMU) sensors available is insufficient to have full observability of the grid state. Proposed analysis uses the Kullback Leibler (KL) divergence between the distributions corresponding to different fault location hypotheses associated with the observation model. This analysis shows that the most likely locations are concentrated in clusters of buses more tightly connected to the actual fault site akin to graph communities. Consequently, a PMU placement strategy is derived that achieves a near-optimal resolution for localizing faults for a given number of sensors. The problem is also analyzed from the perspective of sampling a graph signal, and how the placement of the PMUs i.e. the spatial sampling pattern and the topological characteristic of the grid affect the ability to successfully localize faults. To highlight the superior performance of presented fault localization and placement algorithms, the proposed strategy is applied to a modified IEEE 34, IEEE-123 bus test cases and to data from a real distribution grid. Additionally, the detection of cyber-physical attacks is also examined where PMU data and relevant Supervisory Control and Data Acquisition (SCADA) network traffic information are compared to determine if a network breach has affected the integrity of the system information and/or operations

The Total Acquisition Number of the Randomly Weighted Path

There exists a significant body of work on determining the acquisition number $a_t(G)$ of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number $E(a_t(P_n))$ of the $n$-path when $n$ distinguishable "units" of integral weight, or chips, are randomly distributed across its vertices between $0.242n$ and $0.375n$. With computer support, we improve it by showing that $E(a_t(P_n))$ lies between $0.29523n$ and $0.29576n$. We then use subadditivity to show that the limiting ratio $\lim E(a_t(P_n))/n$ exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set.Comment: 19 page

Applications of the DOE/NASA wind turbine engineering information system

A statistical analysis of data obtained from the Technology and Engineering Information Systems was made. The systems analyzed consist of the following elements: (1) sensors which measure critical parameters (e.g., wind speed and direction, output power, blade loads and component vibrations); (2) remote multiplexing units (RMUs) on each wind turbine which frequency-modulate, multiplex and transmit sensor outputs; (3) on-site instrumentation to record, process and display the sensor output; and (4) statistical analysis of data. Two examples of the capabilities of these systems are presented. The first illustrates the standardized format for application of statistical analysis to each directly measured parameter. The second shows the use of a model to estimate the variability of the rotor thrust loading, which is a derived parameter

Internal combustion engine sensor network analysis using graph modeling

In recent years there has been a rapid development in technologies for smart monitoring applied to many different areas (e.g. building automation, photovoltaic systems, etc.). An intelligent monitoring system employs multiple sensors distributed within a network to extract useful information for decision-making. The management and the analysis of the raw data derived from the sensor network includes a number of specific challenges still unresolved, related to the different communication standards, the heterogeneous structure and the huge volume of data. In this paper we propose to apply a method based on complex network theory, to evaluate the performance of an Internal Combustion Engine. Data are gathered from the OBD sensor subset and from the emission analyzer. The method provides for the graph modeling of the sensor network, where the nodes are represented by the sensors and the edge are evaluated with non-linear statistical correlation functions applied to the time series pairs. The resulting functional graph is then analyzed with the topological metrics of the network, to define characteristic proprieties representing useful indicator for the maintenance and diagnosis

Two-Stage Code Acquisition Employing Search Space Reduction and Iterative Detection in the DS-UWB Downlink

Abstract—In this paper we propose and investigate an iterative code acquisition scheme assisted by both search space reduction and iterative Massage Passing (MP), which was designed for the Direct Sequence-Ultra WideBand (DS-UWB) DownLink (DL). The performance of this iterative code acquisition scheme is analysed in terms of both the correct detection probability and the achievable Mean Acquisition Time (MAT). We propose an improved criterion for designing the iterative MP based twostage acquisition regime. Our proposed scheme is capable of reducing the MAT by several orders of magnitude compared to the benchmark scenarios, when considering the employment of long PseudoNoise (PN) codes suitable for a variety of applications

Development and implementation of a LabVIEW based SCADA system for a meshed multi-terminal VSC-HVDC grid scaled platform

This project is oriented to the development of a Supervisory, Control and Data Acquisition (SCADA) software to control and supervise electrical variables from a scaled platform that represents a meshed HVDC grid employing National Instruments hardware and LabVIEW logic environment. The objective is to obtain real time visualization of DC and AC electrical variables and a lossless data stream acquisition. The acquisition system hardware elements have been configured, tested and installed on the grid platform. The system is composed of three chassis, each inside of a VSC terminal cabinet, with integrated Field-Programmable Gate Arrays (FPGAs), one of them connected via PCI bus to a local processor and the rest too via Ethernet through a switch. Analogical acquisition modules were A/D conversion takes place are inserted into the chassis. A personal computer is used as host, screen terminal and storing space. There are two main access modes to the FPGAs through the real time system. It has been implemented a Scan mode VI to monitor all the grid DC signals and a faster FPGA access mode VI to monitor one converter AC and DC values. The FPGA application consists of two tasks running at different rates and a FIFO has been implemented to communicate between them without data loss. Multiple structures have been tested on the grid platform and evaluated, ensuring the compliance of previously established specifications, such as sampling and scanning rate, screen refreshment or possible data loss. Additionally a turbine emulator was implemented and tested in Labview for further testing