Faculty of Science https://hdl.handle.net/10646/1380 2021-04-14T07:06:11Z An Upper Bound on the Radius of a 3-Vertex-Connected C4-Free Graph https://hdl.handle.net/10646/3964 An Upper Bound on the Radius of a 3-Vertex-Connected C4-Free Graph Fundikwa, Blessings T.; Mazorodze, Jaya P.; Mukwembi, Simon Let G=(V,E) be a finite, connected, undirected graph with vertex set V and edge set E. -e distance dG (u,v) between two vertices u, v of G is the length of a shortest u-v path in G.-e eccentricity ec(v)of a vertex v∈V is the maximum distance between v and any other vertex in G. -e value of the minimum eccentricity of the vertices of G is called the radius of G denoted by rad(G). -e degree deg(v)of a vertex v of G is the number of edges incident with v. -e minimum degree δ(G)is the minimum of the degrees of vertices in G.-e open neighbourhood N(v)of a vertex v is the set of all vertices of G adjacent to v. -e closed neighbourhood N[v]of v is the set N(v)∪v{ }. A graph is triangle-free if it does not contain C3 as a subgraph and C4−free if it does not contain C4 as a subgraph. For notions not defined, here we refer the reader to . The results in this paper are part of the first author’s MPhilSc thesis. 2020-08-04T00:00:00Z Upper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge Connectivity https://hdl.handle.net/10646/3963 Upper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge Connectivity Fundikwa, Blessings T.; Mazorodze, Jaya P.; Mukwembi, Simon Graph theory is used to study the mathematical structures of pairwise relations among objects. Mathematically, a pair G=(V,E) is a crisp graph, where V is a nonempty set and E is a relation on V. -e order of a graph G is the number of vertices of G and is denoted by|V|=n. -e size of G,denoted by|E|=m, is the number of edges of G. -e distance,d G(u,v), between two vertices u,v of G is the length of a shortest u-v path in G. -e eccentricity, ec G(v), of a vertex v∈V is the maximum distance between v and any other vertex in G. -e maximum distance among all pairs of vertices , also known as the value of the maximum ec-centricity of the vertices of G, is called the diameter of G denoted by diam(G). -e degree, deg(v), of a vertex v of G is the number of edges incident with v. -e minimum degree,δ(G)=δ, of G is the minimum of the degrees of vertices in G. -e open neighborhood,N(v), of a vertex v is simply the set containing all the vertices adjacent to v. -e closed neighborhood, N[v], of a vertex v is simply the set containing the vertex v itself and all the vertices adjacent to v.We denote by E(V1,V2)the set {e=xy|x∈V1,y∈V2}of edges with one end inV1and the other end inV2.-e edge connectivity, λ(G)=λ, of G is the minimum number of edges whose deletion from G results in a disconnected or trivial graph. A complete graph,Kn, is a graph in which every vertex is adjacent to every other vertex. -e most likely antonym for a complete graph is a null graph,Nn, which is a graph containing only vertices and no edges. A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge in G connects a vertex in U to a vertex in V; furthermore, no two vertices in the same set are adjacent to each other. A graph is triangle-free if it does not contain C3 as a subgraphandC4-free if it does not containC4as a subgraph. It is important to observe from the above definitions that every bipartite graph is triangle free, but there are some triangle-free graphs which are not bipartite, for example, a cycle graph with five vertices(C5). For notions not defined here,we refer the reader to . 2020-09-03T00:00:00Z Spanning paths in graphs https://hdl.handle.net/10646/3962 Spanning paths in graphs Mafuta, Phillip; Mukwembi, Simon; Munyira, Sheunesu The Conjecture, Graffiti.pc 190, of the computer program Graffiti.pc, instructed by DeLavi˜na, state that every simple connected graph G with minimum degree δ and leaf number L(G) such that δ ≥ 1 2 (L(G) + 1), is traceable. Here, we prove a sufficient condition for a graph to be traceable based on minimum degree and leaf number, by settling completely, the Conjecture Graffiti. pc 190. We construct infinite graphs to show that our results are best in a sense. All graphs considered are simple. That is, they neither have loops nor multiple edges. 2018-08-29T00:00:00Z Modeling the transmission dynamics of brucellosis https://hdl.handle.net/10646/3918 Modeling the transmission dynamics of brucellosis Paride, Paride Oresto Lolika Brucellosis, a neglected zoonotic disease remains a major public health problem world over. It affects domesticated animals, wildlife and humans. With large pastoral communities, and demand for meat and livestock production to double by 2050, brucellosis poses a major threat to the public health and economic growth of several developing nations whose economies rely heavily on agricultural exports. Since human-to-human transmission of the disease is rare the ultimate management of human brucellosis can be achieved through effective control of brucellosis in animal population. Hence there is need to gain a better and more comprehensive understanding of effective ways to control the disease in animal populations. Mathematical modeling, analysis and simulation offer a useful means to understand the transmission and spread of brucellosis so that effective disease control measures could be designed. In this dissertation, five epidemiological models that seek to evaluate the role of intervention strategies on the transmission dynamics of brucellosis in animal population have been studied. Firstly, a non-autonomous model that focuses on evaluating the impact of animal vaccination and environmental decontamination in a periodic environment, is introduced. Secondly, a modeling framework that seeks to improve our quantitative understanding of the influence of chronic brucellosis and culling control on brucellosis dynamics in periodic and non-periodic environments, is considered. Thirdly, a deterministic brucellosis model that incorporates heterogeneity and seasonality is studied. Fourthly, we evaluated the effects of short-term animal movements on the transmission dynamics of brucellosis through a two-patch model. Finally a model that incorporates two discrete delays and culling of infected animals displaying signs of brucellosis infection is proposed and analysed. All the proposed models incorporate relevant biological and ecological factors as well as possible disease intervention strategies. Epidemic and endemic analysis of the models have been performed, with a focus on the threshold dynamics characterized by the basic reproduction numbers. In addition, numerical simulation results are presented to demonstrate the analytical findings. A brief summary of the main results of the thesis and an outline of some possible future research directions rounds up the thesis. 2019-10-01T00:00:00Z