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Finding a common language in describing and interpreting multivariate data associated with rehabilitation and disturbance ecology, has became a major challenge. The main objective of this study is to find and evaluate mathematical models to describe ecosystem change based on selected indicators of change. Existing data from a previous rehabilitation project on Hendrina Power Station (Mpumalanga, South Africa) was used as a database for this study and this study aims to report on the development of models concentrating on radar graphs and a model based on matrix mathematics. The main groups of organisms selected for the construction of models, were vegetation, soil mesofauna and ant species. The datasets were limited to some indicative species and their mean abundances were determined. The grids that were used were randomly chosen and the models were constructed. Radar graphs were constructed to model the suite of species identified, through a sensitivity analysis, to indicate possible rehabilitation success over time and was applied to the different rehabilitation ages. The surface areas under the radar graphs were determined and compared for the different rehabilitation ages in the same year of survey. Correlation graphs were drawn between the surface area and the rehabilitation ages. These graphs did not indicate much relevance in indicating rehabilitation success, but the radar graphs proved to be good indicators of change in abundance of the selected species over time. iv The vegetation species, Eragrostis curvula, was the only species that showed a strong significant positive relationship with rehabilitation age and could be considered a good rehabilitation species and indicator of rehabilitation success. After the evaluation of this model, Eragrostis curvula, and two additional ant species, Tetramorium setigerum and Lepisiota laevis, were added. These species that were added, showed an increase in abundance over time, as found in a previous study. These radar graphs also did not indicate much relevance and it can be concluded that the radar graphs can only be used for a visual representation of the changes in abundance of the relevant species over time. This study also refers to a matrix model. This model focused on the interactions between the different variables selected. The percentage carbon in the soil were also added to the list of species. Model fitting graphs were constructed and correlations were drawn between the species that had significant values in the interaction table. This model could be useful for future studies, but more data and replication is necessary, over a longer period of time. This will serve to eliminate possible shortcomings of the model. / Thesis (M. Environmental Science (Biodiversity and Conservation Biology))--North-West University, Potchefstroom Campus, 2007.

We will discuss three ways to bound the chromatic number on a Cayley graph. 1. If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. 2. We will prove a general statement that all vertex-transitive maximal triangle-free graphs on <i>n</i> vertices with valency greater than <i>n</i>/3 are 3-colourable. Since Cayley graphs are vertex-transitive, the bound of general graphs also applies to Cayley graphs. 3. Since Cayley graphs for abelian groups arise from vector spaces, we can view the connection set as a set of points in a projective geometry. We will give a characterization of all large complete caps, from which we derive that all maximal triangle-free cubelike graphs on 2ⁿ vertices and valency greater than 2ⁿ/4 are either bipartite or 4-colourable.

The chemical master equation (CME) represents the accepted stochastic description of chemical reaction kinetics in mesoscopic systems. As its exact solution – which gives the corresponding probability density function – is possible only in very simple cases, there is a clear need for approximation techniques. Here, we propose a novel perturbative three-step approach which draws heavily on graph theory: (i) we expand the eigenvalues of the transition state matrix in the CME as a series in a non-dimensional parameter that depends on the reaction rates and the reaction volume; (ii) we derive an analogous series for the corresponding eigenvectors via a graph-based algorithm; (iii) we combine the resulting expansions into an approximate solution to the CME. We illustrate our approach by applying it to a reversible dimerization reaction; then, we formulate a set of conditions, which ensure its applicability to more general reaction networks. We follow attempting to apply the results to a more complicated system, namely push-pull, but the problem reveals too complex for a complete solution. Finally, we discuss the limitations of the methodology.

Normal coloring - an equivalent version of Petersen coloring - is a special proper 5-edge-coloring of cubic graphs. Every edge in a normally colored graph is normal, i.e. it uses together with its four neighbours either only three colors or all five colors. Jaeger conjectured that every bridgeless cubic graph has a normal coloring. This conjecture, if true, imply for example Cycle double cover conjecture. Here we solve a weakened version of Jaeger's problem. We are looking for a proper 5-edge-coloring such that at least a part of the edges is normal. We show a coloring of generalized prisms with two thirds of the edges normal and a coloring of graphs without short cycles with almost half of the edges normal. Then we propose a new approach to normal coloring - chains. We use chains to prove that there cannot be only one single mistake in an almost normally colored graph. We also prove some statements about cuts in a normally colored graph which also follow from nowhere-zero Petersen flow. Finally, we examine a four-cycle in a normally colored graph. 1

Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like G_t=(V,E_t), where E_(t-1) in E_t, t=1,... ,T; E_0=(). In this article algorithms are provided to track all maximal cliques in a fully dynamic graph. It is naturally to raise the question about the maximum clique, having all maximal cliques. Therefore this article discusses potentials and drawbacks for this problem as well. (author's abstract) / Series: Working Papers on Information Systems, Information Business and Operations

Un graphe est un ensemble de noeuds, ensemble de liens reliant des paires de noeuds. Avec la quantité accumulée de données collectées, il existe un intérêt croissant pour la compréhension des structures et du comportement de très grands graphes. Néanmoins, l’augmentation rapide de la taille des grands graphes rend l’étude de tous les graphes de moins en moins efficace. Ainsi, il existe une demande impérieuse pour des méthodes plus efficaces pour étudier de grands graphes sans nécessiter la connaissance de tous les graphes. Une méthode prometteuse pour comprendre le comportement de grands graphes consiste à exploiter des propriétés spécifiques de structures locales, telles que la taille des grappes ou la présence locale d’un motif spécifique, c’est-à-dire un graphe donné (généralement petit). Un exemple classique de la théorie des graphes (cas avérés de la conjecture d'Erdos-Hajnal) est que, si un graphe de grande taille ne contient pas de motif spécifique, il doit alors avoir un ensemble de noeuds liés par paires ou non liés, de taille exponentiellement plus grande que prévue. Cette thèse abordera certains aspects de deux questions fondamentales de la théorie des graphes concernant la présence, en abondance ou à peine, d’un motif donné dans un grand graphe : - Le grand graphe peut-il être partitionné en copies du motif ? - Le grand graphe contient-il une copie du motif ? Nous discuterons de certaines des conjectures les plus connues de la théorie des graphes sur ce sujet: les conjectures de Tutte sur les flots dans les graphes et la conjecture d'Erdos-Hajnal mentionnée ci-dessus, et présenterons des preuves pour plusieurs conjectures connexes - y compris la conjecture de Barát-Thomassen, une conjecture de Haggkvist et Krissell, un cas particulier de la conjecture de Jaeger-Linial-Payan-Tarsi, une conjecture de Berger et al, et une autre d'Albouker et al. / A graph is a set of nodes, together links connecting pairs of nodes. With the accumulating amount of data collected, there is a growing interest in understanding the structures and behavior of very large graphs. Nevertheless, the rapid increasing in size of large graphs makes studying the entire graphs becomes less and less efficient. Thus, there is a compelling demand for more effective methods to study large graphs without requiring the knowledge of the graphs in whole. One promising method to understand the behavior of large graphs is via exploiting specific properties of local structures, such as the size of clusters or the presence locally of some specific pattern, i.e. a given (usually small) graph. A classical example from Graph Theory (proven cases of the Erdos-Hajnal conjecture) is that if a large graph does not contain some specific pattern, then it must have a set of nodes pairwise linked or not linked of size exponentially larger than expected. This thesis will address some aspects of two fundamental questions in Graph Theory about the presence, abundantly or scarcely, of a given pattern in some large graph: - Can the large graph be partitioned into copies of the pattern? - Does the large graph contain any copy of the pattern?We will discuss some of the most well-known conjectures in Graph Theory on this topic: the Tutte's flow conjectures on flows in graphs and the Erdos-Hajnal conjecture mentioned above, and present proofs for several related conjectures -- including the Barát-Thomassen conjecture, a conjecture of Haggkvist and Krissell, a special case of Jaeger-Linial-Payan-Tarsi's conjecture, a conjecture of Berger et al, and another one by Albouker et al.