Browsing by Author "Mayala, Roger Mbonga."

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The eigen-chromatic ratio of classes of graphs : molecular stability, asymptotes and area.
(2017) Mayala, Roger Mbonga.; Winter, Paul August.; Namayanja, Proscovia.
This dissertation involves combining the two concepts of energy and the chromatic number of classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this ratio is the importance of its asymptotic convergence in applications, as well as the idea of area involving the Rieman integral of this ratio, when it is a function of the order n of the graph G belonging to a class of graphs. The energy of a graph G, is the sum of the absolute values of the eigenvalues associated with the adjacency matrix of G, and its importance has found its way into many areas of research in graph theory. The chromatic number of a graph G, is the least number of colours required to colour the vertices of the graph, so that no two adjacent vertices receive the same colour. The importance of ratios in graph theory is evident by the vast amount of research articles: Expanders, The central ratio of a graph, Eigen-pair ratio of classes of graphs , Independence and Hall ratios, Tree-cover ratio of graphs, Eigen-energy formation ratio, The eigen-complete difference ratio, The chromatic-cover ratio and "Graph theory and calculus: ratios of classes of graphs". We combine the two concepts of energy and chromatic number (which involves the order n of the graph G) in a ratio, called the eigen-chromatic ratio of a graph. The chromatic number associated with the molecular graph (the atoms are vertices and edges are bonds between the atoms) would involve the partitioning of the atoms into the smallest number of sets of like atoms so that like atoms are not bonded. This ratio would allow for the investigation of the effect of the energy on the atomic partition, when a large number of atoms are involved. The complete graph is associated with the value 1 2 when the eigen-chromatic ratio is investigated when a large number of atoms are involved; this has allowed for the investigation of molecular stability associated with the idea of hypo/hyper energetic graphs. Attaching the average degree to the Riemann integral of this ratio (as a function of n) would result in an area analogue for investigation. Once the ratio is defned the objective is to find the eigen-chromatic ratio of various well known classes of graphs such as the complete graph, bipartite graphs, star graphs with rays of length two, wheels, paths, cycles, dual star graphs, lollipop graphs and caterpillar graphs. Once the ratio of each class of graph are determined the asymptote and area of this ratio are determined and conclusions and conjectures inferred.
Role of Weyl tensor and spacetime shear in relativistic fluids.
(2021) Mayala, Roger Mbonga.; Maharaj, Sunil Dutt.; Goswami, Rituparno.
The main gravitational theory in which we develop this work is general relativity. We study the role of the Weyl tensor in general relativistic fluid motion including the e↵ects of spacetime shear. Firstly we consider conformally flat perturbations on the Friedmann Lemaitre RobertsonWalker (FLRW) spacetime containing a general matter field. Working with the linearised field equations, we find some important geometrical properties of matter shear and vorticity, and show how they interact with the thermodynamic quantities in the absence of any free gravity powered by the Weyl curvature. We demonstrate that the matter shear obeys a transverse traceless tensor wave equation and the vorticity obeys a vector wave equation in this linearised regime. These shear and vorticity waves replace the gravitational waves in the sense that they causally carry information about local change in the curvature of these spacetimes. We also study the heat transport equation in this case, and show how this varies from the Newtonian case. Secondly we show that a general but shear-free perturbation of homogeneous and isotropic universes are necessarily silent, without any gravitational waves. We prove this in two steps. First, we establish that a shear-free perturbation of these universes are acceleration-free and the fluid flow geodesics of the background universe map onto themselves in the perturbed universe. This e↵ect then decouples the evolution equations of the electric and magnetic part of the Weyl tensor in the perturbed spacetimes and the magnetic part no longer contains any tensor modes. Although the electric part, that drives the tidal forces, does have tensor modes sourced by the anisotropic stress, these modes have homogeneous oscillations at every point on a time slice without any wave propagation. This analysis shows the critical role of the shear tensor in generating cosmological gravitational waves.