Walks and games on graphs

Herrman, Rebekah Ph.D. The University of Memphis, May 2020. Walks and Games on Graphs. Major Professor: B\\u27ela Bollob\\u27as, Ph.D.Chapter 1 is joint work with Dr. Travis Humble and appears in the journal Physical Review A. In this work, we consider continuous-time quantum walks on dynamic graphs. Continuous-time quantum walks have been well studied on graphs that do not change as a function of time. We offer a mathematical formulation for how to express continuous-time quantum walks on graphs that can change in time, find a universal set of walks that can perform any operation, and use them to simulate basic quantum circuits. This work was supported in part by the Department of Energy Student Undergraduate Laboratory Internship and the National Science Foundation Mathematical Sciences Graduate Internship programs.The $(t,r)$ broadcast domination number of a graph $G$, $\gamma_{t,r}(G)$, is a generalization of the domination number of a graph. In Chapter 2, we consider the $(t,r)$ broadcast domination number on graphs, specifically powers of cycles, powers of paths, and infinite grids. This work is joint with Peter van Hintum and has been submitted to the journal Discrete Applied Mathematics.Bridge-burning cops and robbers is a variant of the cops and robbers game on graphs in which the robber removes an edge from the graph once it is traversed. In Chapter 3, we study the maximum time it takes the cops to capture the robber in this variant. This is joint with Peter van Hintum and Dr. Stephen Smith.In Chapter 4, we study a variant of the chip-firing game called the \emph{diffusion game}. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever we have a graph on $n$ vertices and an initial allocation with at least $f(n)$ chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that $f(n)=n-2$ is the best possible bound. We also consider the existence of a similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the graph. This work is joint with Andrew Carlotti and has been submitted to the journal Discrete Mathematics.In Chapter 5, we consider the eternal game chromatic number of random graphs. The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza \cite{klostermeyer}, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this chapter, we show that with high probability $\chi_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \N$. This work is joint with Vojt\u{e}ch Dvo\u{r}\\u27ak and Peter van Hintum, and has been submitted to the European Journal of Combinatorics

View generated database

This document represents the final report for the View Generated Database (VGD) project, NAS7-1066. It documents the work done on the project up to the point at which all project work was terminated due to lack of project funds. The VGD was to provide the capability to accurately represent any real-world object or scene as a computer model. Such models include both an accurate spatial/geometric representation of surfaces of the object or scene, as well as any surface detail present on the object. Applications of such models are numerous, including acquisition and maintenance of work models for tele-autonomous systems, generation of accurate 3-D geometric/photometric models for various 3-D vision systems, and graphical models for realistic rendering of 3-D scenes via computer graphics

Connectivity properties of some transformation graphs

Many combinatorial problems can be formulated as \can we transform configuration 1 into configuration 2 if certain transformations are allowed?" In order to study such questions, we introduce a so-called transformation graph. This graph has the set of all possible configurations as its vertex set, and there is an edge between two configurations if one configuration can be obtained from the other by one of the allowed transformations. Then a question like \can we go from one configuration to another one" becomes a question about connectivity properties of transformation graphs. In this thesis, we study the following types of transformation graphs in particular: Labelled Token Graphs: Here configurations are arrangements of labelled tokens on a given graph, and we can go from one arrangement to another one by moving one token at a time along an edge of the given graph. We classify all cases when labelled token graphs are connected, and classify all pairs of arrangements that are in the same component. We also look at the problem how hard it is to determine the length of the shortest path between two arrangements. Strong k-Colour Graphs: For this transformation graph, the configurations are the proper vertex-colourings of a given graph with k colours, in which all k colours are actually used. We call such a colouring a strong k-colouring. We study the problem when we can transform any strong k-colouring into any other one by recolouring one vertex at a time, always maintaining a strong k-colouring. For certain classes of graphs, we can completely determine when the transformation graph of strong k-colourings is connected

Topics in gravitational-wave astronomy: Theoretical studies, source modelling and statistical methods

Astronomy with gravitational-wave observations is now a reality. Much of the theoretical research in this field falls under three broad themes: the mathematical description and physical understanding of gravitational radiation and its effects; the construction of accurate and computationally efficient waveform models for astrophysical sources; and the improved statistical analysis of noisy data from interferometric detectors, so as to extract and characterise source signals. The doctoral thesis presented in this dissertation is an investigation of various topics across these themes. Under the first theme, we examine the direct interaction between gravitational waves and electromagnetic fields in a self-contained theoretical study; this is done with a view to understanding the observational implications for highly energetic astrophysical events that radiate in both the gravitational and electromagnetic sectors. We then delve into the second theme of source modelling by developing and implementing an improved waveform model for the extreme-mass-ratio inspirals of stellar-mass compact objects into supermassive black holes, which are an important class of source for future space-based detectors such as the Laser Interferometer Space Antenna. Two separate topics are explored under the third theme of data analysis. We begin with the procedure of searching for gravitational-wave signals in detector data, and propose several combinatorial compression schemes for the large banks of waveform templates that are matched against putative signals, before studying the usefulness of these schemes for accelerating searches. After a gravitational-wave source is detected, the follow-up process is to measure its parameters in detail from the data; this is addressed as we apply the machine-learning technique of Gaussian process regression to gravitational-wave data analysis, and in particular to the formidable problem of parameter estimation for extreme-mass-ratio inspirals

Application of quantum walks on graph structures to quantum computing

Quantum computation is a new computational paradigm which can provide fundamentally faster computation than in the classical regime. This is dependent on finding efficient quantum algorithms for problems of practical interest. One of the most successful tools in developing new quantum algorithms is the quantum walk. In this thesis, we explore two applications of the discrete time quantum walk. In addition, we introduce an experimental scheme for generating cluster states, a universal resource for quantum computation. We give an explicit construction which provides a link between the circuit model of quantum computation, and a graph structure on which the discrete time quantum walk traverses, performing the same computation. We implement a universal gate set, proving the discrete time quantum walk is universal for quantum computation, thus confirming any quantum algorithm can be recast as a quantum walk algorithm. In addition, we study factors affecting the efficiency of the quantum walk search algorithm. Although there is a strong dependence on the spatial dimension of the structure being searched, we find secondary dependencies on other factors including the connectivity and disorder (symmetry). Fairly intuitively, as the connectivity increases, the efficiency of the algorithm increases, as the walker can coalesce on the marked state with higher probability in a quicker time. In addition, we find as disorder in the system increases, the algorithm can maintain the quantum speed up for a certain level of disorder before gradually reverting to the classical run time. Finally, we give an abstract scheme for generating cluster states. We see a linear scaling, better than many schemes, as doubling the size of the generating grid in our scheme produces a cluster state which is double the depth. Our scheme is able to create other interesting topologies of entangled states, including the unit cell for topological error correcting schemes

Technology 2000, volume 1

The purpose of the conference was to increase awareness of existing NASA developed technologies that are available for immediate use in the development of new products and processes, and to lay the groundwork for the effective utilization of emerging technologies. There were sessions on the following: Computer technology and software engineering; Human factors engineering and life sciences; Information and data management; Material sciences; Manufacturing and fabrication technology; Power, energy, and control systems; Robotics; Sensors and measurement technology; Artificial intelligence; Environmental technology; Optics and communications; and Superconductivity

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..