/library/oar/handle/123456789/57054 Sat, 15 Nov 2025 11:31:06 GMT 2025-11-15T11:31:06Z /library/oar/handle/123456789/65259 Title: Connectivity parameters of the hypercube graph and its variants Abstract: Classical connectivity (edge-connectivity) measures study the minimum number of vertices (or edges) that need to be deleted to disconnect a graph. A network is more fault-tolerant if the number of vertices (or edges) that need to be deleted to disconnect the graph is high. In this dissertation we will focus on the deletion of vertices in order to disconnect graphs. A vertex cut S of a graph G is a set of vertices of G whose deletion results in a disconnected graph or leaves an isolated vertex. The minimum size over all vertex cuts of G is the connectivity κ(G) of the graph G. Many have argued that the connectivity of a graph does not necessarily measure in an accurate way how fault-tolerant a graph would be following the deletion of a set of vertices. This is because it assumes that all the vertices adjacent to any vertex of the graph can fail simultaneously, a particularly remote instance in large–scale processing systems. Thus, other types of connectivity parameters that impose some restrictions on the components of the remaining graph have recently received much attention; a notion proposed by Harary in 1983. Given a graph G and some graph theoretical property P, the size of the smallest set S of vertices, if such a set exists, such that the graph G − S is disconnected and every remaining component has property P is the conditional connectivity of G. Although, theoretically, any property P can be chosen, the most popular choices of P are those having practical applications. One such property that can be chosen is that each of the remaining components has more than h vertices, for some positive integer h. This notion is known as the h–extraconnectivity κh(G) of a graph G. Thus, for h = 0, we have κh(G) = κ(G). Harary proposed the study of the conditional connectivity of some interesting families of graphs, such as complete multipartite graphs, the generalized Petersen graphs and the hypercube graphs. The latter family provides, in fact, a fundamental model for computer networks and these graphs have been extensively used due to their various properties, such as regularity and recursive structure. For any positive integer n, the n-dimensional hypercube is a graph Qn whose vertex set is made up of all possible n-bit binary strings, and two vertices are connected by an edge whenever their binary string representation differs in only one bit position. Thus, Qn has 2n vertices and n2n−1 edges, is regular of degree n and has connectivity equal to n. A number of variants of the n-dimensional hypercubes have been introduced with the aim of further enhancing their properties. Various basic graph-theoretic parameters have been established for these new hypercubes, including the girth, diameter and connectivity. In this dissertation we survey the existing literature on the connectivity and the h-extraconnectivity of hypercube graphs, starting from the work of Yang and Meng on the (standard) n-dimensional hypercube, and proceeding to a number of hypercube variants. The aim of this research is to present and analyse the main results on these two connectivity parameters, and to establish the values of h for which κh is known. Description: M.SC.MATHS Wed, 01 Jan 2020 00:00:00 GMT /library/oar/handle/123456789/65259 2020-01-01T00:00:00Z /library/oar/handle/123456789/63225 Title: Dynamical dark energy models Abstract: Ever since it was discovered that the Universe is expanding at an accelerated rate, cosmologists have been searching for an explanation in the form of dark energy: a mechanism or physical component capable of producing this effect. The study presented here focuses on dynamical dark energy models, in which the observed acceleration arises as a result of either a cosmic component whose pressure is negative, or as a modification to the General Relativistic description of the geometry of the space-time manifold. The word ‘dynamical’ sets these models apart from the ΛCDM cosmology, in which the density of dark energy remains constant as the Universe expands. Many works in the literature are based on the premise of a spatially flat Universe, and indeed this is what observational data appears to point to in a ΛCDM framework. The question naturally arises, however, whether the assumption of flatness continues to hold in the case of dynamical dark energy models, especially since spatial curvature is often correlated with dark energy parameters, and so any wrong assumptions about it could greatly distort our understanding of dark energy. The aim of this thesis is precisely to look for an answer to that question. For the first part of the study, dark energy is modelled as a scalar field that can either be minimally or non-minimally coupled to the Ricci scalar, and a number of exact solutions to the cosmological field equations are presented. Each corresponds to a particular geometry – flat, open or closed. In the next part, analytical methods are combined with numerical techniques to analyse several models from the literature, chosen for their ability to represent the complete cosmic history. The aim is to investigate how spatial curvature influences the main features of the evolution. Initially, the Universe is assumed to consist of a Vander Waalsfluid, but this alone cannot provide an explanation for the acceleration at late times, despite the fact that it accounts for the inflationary and matter-dominated epochs. Hence, dark energy is introduced as Quintessence, a Chaplygin gas or dynamical vacuum energy. It turns out that the transition from the inflationary epoch to the matter-dominated one would occur first for the open universe, and last for the closed one. The onset of late-time acceleration would also take place in this order. Furthermore, positive curvature is found to enhance inflationary acceleration and the deceleration that follows. Among the fluid characteristics considered are the customary proportionality between energy density and pressure, and bulk viscosity. The effects of spatial curvature on cosmic evolution are then investigated in the context of the generalised running vacuum model (GRVM) and its sub-cases. In the GRVM, the cosmological constant is replaced by a function of the Hubble parameter and its time derivative: Λ(H) = A + BH2 + C ˙ H (A, B and C being constants). Two parameter models are obtained by setting B or C equal to zero. The main goal is to find out if the models best describe observations when one assumes spatial flatness, or if the presence of curvature improves the fit. This is accomplished via a Markov Chain Monte Carlo (MCMC) analysis. The data set used comprises measurements of observables related to Type-Ia supernovae, cosmic clocks, baryon acoustic oscillations, the cosmic microwave background and redshift-space distortions (RSDs). Since it is well known that the data itself (rather than just the particular model) plays an important part in determining whether curvature is ruled out, the chapter draws comparisonsbetweentheconstraintsobtainedinvariousscenarios,suchaswhenRSD measurements are excluded (in contrast to when the full data set is employed). The lack of consensus within the scientific community about the value of the Hubble constant(H0) is also taken into account. Two different values of H0 from the literature are introduced and their effects on the results are investigated. In the last part, the focus is shifted to an alternative theory of gravity – namely, f(R) gravity, constructed by generalising the Ricci scalar in the Einstein-Hilbert action to a function thereof. Four f(R) models are considered, all of which appear to be compatible with Solar-System and cosmological constraints: the Hu-Sawicki, Starobinsky, Exponential and Tsujikawa models. The idea is to see whether these models are able to accommodate non-zero spatial curvature (while still being consistent with cosmological observations). Since they all reduce to ΛCD Mathigh redshifts, any differences from ΛCDM are most likely to emerge at the level of perturbations. Therefore, the perturbation equations (in the sub-Hubble, quasi-static regime) are derived and incorporated into the analysis, which is again carried out using MCMC sampling techniques. Given that matter density perturbations are scale-dependent in f(R) gravity, the results obtained for different values of k† (the comoving wave number) are compared. Description: PH.D.MATHS Wed, 01 Jan 2020 00:00:00 GMT /library/oar/handle/123456789/63225 2020-01-01T00:00:00Z /library/oar/handle/123456789/57417 Title: Schur rings : some theory and applications to graphical regular representations Abstract: A Schur ring is a specific type of group ring constructed via the partitioning of a group while the Graphical Regular Representation (GRR) of a group 􀀀 is a graph which has an automorphism group isomorphic to 􀀀 which acts regularly on it. In this dissertation we will be looking at some background theory related to these two concepts and use it to produce an original theorem which allows us to quickly produce trivalent GRRs for all dihedral groups Dp where p is a prime number greater than 5. The background theory we will be considering is mostly work done by other authors over recent years although some new proofs by the author are included, especially in situations where we will be looking at special cases of established work and so prefer a proof specific. to the special cases being considered. The original theorem which we will be building up to is the following: If p is a prime number greater than 5 and 3r 􀀀 2s = t mod p then Cay(Dp; fabr; abs; abtg) is a GRR of Dp. The use of Schur rings will be central to proving this theorem. We will also see a few examples of GRRs produced using this theorem as well as construct a table of connecting sets which admit GRRs for all dihedral groups Dp with p prime and greater than a certain value. Description: M.SC.MATHS Wed, 01 Jan 2020 00:00:00 GMT /library/oar/handle/123456789/57417 2020-01-01T00:00:00Z