Electronic Theses and Dissertations (PhDs)

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Chromatic Polynomials and Certain Classes of Graphs
(University of the Witwatersrand, Johannesburg, 2023) Maphakela, Lesiba Joseph; Mphako-Banda, G.
The chromatic polynomial of a graph has been widely studied in the literature. The focus of this research is on exploring the chromatic polynomial of specific graphs that result from the application of a join operation. The chromatic polynomial of a graph can be expressed in various forms; power form, tree form, factorial form and cycle form. The expressions in various forms, such as power form, tree form, and factorial form, have been subject to comprehensive investigation. However, it should be noted that the cycle form presents relative gaps that necessitate further exploration. This work builds upon the existing literature by engaging in a discussion of the coefficients of the chromatic polynomial of a graph expressed in cycle form. To achieve this objective, we commence by presenting the general formula of the chromatic polynomial in cycle form. Following this, we introduce an algorithm that computes the chromatic polynomial of a graph in cycle form. Additionally, we outline a method for converting the chromatic polynomial of a graph from its tree form into the cycle form. Furthermore, we determine the values of the first and second terms of the chromatic polynomial in its cycle form. This research also complements the well established knowledge of the chromatic polynomial of graphs resulting from the application of a join operation. Of particular interest, we explore the joins of various classes of graphs, including the join of a null graph, N1 with a graph G, which is known as the vertex join of graph G. Building upon this framework, we extend our analysis to encompass the join of a null graph, N2, with graph G. Similarly, we present results pertaining to the join of a complete graph, Kn, with a graph G. Significantly, we conduct a thorough comparative analysis of the chromatic equivalence class among these derived classes of graphs. Lastly, we discuss the chromatic uniqueness of these derived classes of graphs, alongside introducing variations to these derived graphs by deleting their edges and subgraphs.
A symmetry perspective of third-order polynomial evolution equations
(University of the Witwatersrand, Johannesburg, 2024) Gwaxa, Bongumusa; Jamal, Sameerah
In this thesis, we analyse the full class of ten Fujimoto-Watanabe equations. In particular, these are highly nonlinear third-order and two fifth-order equations. With the aid of computer algebra software such as Mathematica, we calculate symmetries for these equations and we construct their commutator tables. The one dimensional system of optimal subalgebras is obtained via adjoint operators. Finally, we reduce these higher-order partial differential equations into ordinary differential equations, derive their solutions via a power series solution method and show how convergence may be tested. Lastly, we determine some conservation laws
An Essay on Branching Time Logics
(University of the Witwatersrand, Johannesburg, 2024) Marais, Chantel
In this thesis we investigate the Priorian logics of a variety of classes of trees. These classes of trees are divided in to irreflexive and reflexive trees, and each of these has a number of subclasses, for example, dense irreflexive trees, discrete reflexive trees, irreflexive trees with branches isomorphic to the natural numbers, etc. We find finite axiomatisations for the logics of these different classes of trees and show that each logic is sound and strongly / weakly complete with respect to the respective class of trees. The methods use to show completeness vary from adapting some known constructions for specific purposes, including unravelling and bulldozing, building a network step-by-step, filtering through a finite set of formulas, as well as using some new processes, namely refining the filtration and unfolding. Once the logics have been shown to be sound and complete with respect to the different classes of trees, we also show that most of these logics are decidable, using methods that include the finite model property, mosaics and conservative extensions. Lastly, we give a glimpse into the available research on other languages used to study branching time structures, including the Peircean and Ockhamist languages, and languages that include additional modal operators like “since” and “until”
The role of invariants in obtaining exact solutions of differential equations
(University of the Witwatersrand, Johannesburg, 2024) Ahmed, Mogahid Mamoon Abkar; Kara, A.H.
We show here that variational and gauge symmetries have additional appli- cations to the integrability of differential equations. We present a general method to construct first integrals for some classes. In particular, we present a broad class of diffusion type equations, viz., the Fisher Kolmorov and Fitzhugh Nagumo equations, which satisfy the Painlev´e properties of their respective travelling wave forms and solitons. It is then shown how a study of invari- ance properties and conservation laws is used to ‘twice’ reduce the equations to solutions. We further constructing the first integrals of a large class of the well-known second-order Painlev´e equations. In some cases, variational and gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether’s theorem. Generally, it is more convenient to adopt the ‘multiplier’ approach to find the first integrals. The main chapters of this thesis have either been published or submitted for publication in accredited journals. The contents of Chapters 2, 3 and 5 has been published ([54], [55]). All computations were done either by hand or Maple

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