Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table

Exploring player experience and social networks in MOBA Games: The case of League of Legends

A pesar de la popularidad de los juegos de arena de combate multijugador en línea (MOBA en inglés) como League of Legends (LoL), tanto la experiencia de jugador (PE) que proporciona este género relativamente reciente como las redes sociales que se generan a su alrededor siguen, en gran medida, inexplorados. Con el incremento del tiempo que los jugadores dedican a este tipo de juegos competitivos en línea, los impactos positivos y negativos de hacerlo cobran relevancia; es, por lo tanto, importante entender cómo se estructura dicha experiencia para abordar de forma sistemática los mecanismos que desencadenan respuestas de los jugadores. El presente trabajo empieza obteniendo y caracterizando una muestra de jugadores de League of Legends y sigue con el uso de las variables resultantes y de la estructura de las relaciones sociales como entradas para explorar su relación con la experiencia de los jugadores. Al fin y al cabo, la PE es básica para involucrar al jugador y, por lo tanto, es clave para el éxito de cualquier juego digital. Los resultados muestran, entre otros, cómo los jugadores de League of Legends perciben el juego como “justo” para su nivel de competencia en cualquier rango, mientras que su afinidad respecto a los compañeros se ve afectada por la estructura social. La empatía y los sentimientos negativos, no obstante, no parecen verse afectados por la composición del equipo. Entender la experiencia del jugador en League of Legends puede no tan sólo ser útil para mejorar el propio LoL o los juegos de tipo MOBA, sino también para desarrollar juegos más inmersivos a la vez que se mejora su calidad. A medida que los juegos competitivos online se convierten rápidamente en una de las mayores actividades colectivas humanas a nivel global, la investigación sobre la experiencia del jugador adquiere también una importancia crucial

On Self-Dual Quantum Codes, Graphs, and Boolean Functions

A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure

Improving the capacity of radio spectrum: exploration of the acyclic orientations of a graph

The efficient use of radio spectrum depends upon frequency assignment within a telecommunications network. The solution space of the frequency assignment problem is best described by the acyclic orientations of the network. An acyclic orientation Ɵ of a graph (network) G is an orientation of the edges of the graph which does not create any directed cycles. We are primarily interested in how many ways this is possible for a given graph, which is the count of the number of acyclic orientations, a(G). This is just the evaluation of the chromatic polynomial of the graph χ(G; λ) at λ = -1. Calculating (and even approximating) the chromatic polynomial is known to be #P-hard, but it is unknown whether or not the approximation at the value -1 is. There are two key contributions in this thesis. Firstly, we obtain computational results for all graphs with up to 8 vertices. We use the data to make observations on the structure of minimal and maximal graphs, by which we mean graphs with the fewest and greatest number of acyclic orientations respectively, as well as on the distribution of acyclic orientations. Many conjectures on the structure of extremal graphs arise, of which we prove some in the theoretical part of the thesis. Secondly, we present a compression move which is monotonic with respect to the number of acyclic orientations, and with respect to various other parameters in particular cliques. This move gives us a new approach to classifying all minimal graphs. It also enables us to tackle the harder problem of identifying maximal graphs. We show that certain Turán graphs are uniquely maximal (Turán graphs are complete multipartite graphs with all vertex classes as equal as possible), and conjecture that all Turán graphs are maximal. In addition we derive an explicit formula for the number of acyclic orientations of complete bipartite graphs