THE EIGEN-COMPLETE DIFFERENCE RATIO OF CLASSES OF GRAPHS- DOMINATION, ASYMPTOTES AND AREA

The energy of a graph   is related to the sum of Ï€ -electron energy in a molecule represented by a molecular graph, and originated by the  HMO (Hückel molecular orbital) theory.  Advances to this theory have taken place which includes the difference of the energy of graphs and the energy formation difference between a graph and its decomposable parts. Although the complete graph does not have the highest energy of all graphs, it is significant in terms of its easily accessible graph theoretical properties, and has a  high level of connectivity and robustness, for example. In this paper we introduce a ratio, the eigen-complete difference ratio, involving the difference in energy between the complete graph and any other connected graph G, which allows for the investigation of the effect of energy of G with respect to the complete graph when a large number of vertices are involved. This is referred to as the eigen-complete difference domination effect. This domination effect is greatest negatively (positively), for a strongly regular graph (star graphs with rays of length one), respectively, and zero for the lollipop graph. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the eigen-complete difference area aspect of classes of graphs.  We applied these eigen-complete aspects to complements of classes of graphs

An analysis of learners' engagement in mathematical task.

Thesis (M.Ed.) - University of Natal, Durban, 1988.The present project is part of a larger research programme focussed on the analysis of change; one aspect being educational transformation and in particular an emphasis on the explication of the contentless processes (eg. logical operations, reasoning styles, analysis and synthesis) which underlie both learning and teaching at university level. The present project is aimed at an analysis of the teaching-learning dialectic in mathematics courses. This analysis has two major focal points, that is, making explicit the often tacit and mostly inadequate and/or inappropriate rules for engaging in mathematical tasks which the under-prepared learner brings to the teaching-learning situation, and secondly the teaching strategies which may enable these learners to overcome their past (erroneous) knowledge and skills towards the development of effecient, autonomous mathematical problem-solving strategies. In order to remedy inadequate and inappropriate past learning and/or teaching, the present project presents a set of mediational strategies and regulative cues which function both for the benefit of the teacher and the learner in a problematic teaching-learning situation and on the meta and epistemic cognitive levels of information processing. Furthermore, these mediational strategies and regulative cues fall on a kind of interface between contentless processes and the particular content of the teaching-learning dialectic of mathematics in particular, as well as between the ideal components of any instructional process and the particular needs and demands of under-prepared learners engaged in mathematical tasks

The Eigen-chromatic Ratio of Classes of Graphs: Asymptotes, Areas and Molecular Stability

In this paper, we present a new ratio associated with classes of graphs, called the eigen-chromatic ratio, by combining the two graph theoretical concepts of energy and chromatic number. The energy of a graph, the sum of the absolute values of the eigenvalues of the adjacency matrix of a graph, arose historically as a result of the energy of the benzene ring being identical to that of the sum of the absolute values of the eigenvalues of the adjacency matrix of the cycle graph on n vertices (see [18]). The chromatic number of a graph is the smallest number of colour classes that we can partition the vertices of a graph such that each edge of the graph has ends that do not belong to the same colour class, and applications to the real world abound (see [30]). Applying this idea to molecular graph theory, for example, the water molecule would have its two hydrogen atoms coloured with the same colour different to that of the oxygen molecule. Ratios involving graph theoretical concepts form a large subset of graph theoretical research (see [3], [16], [48]). The eigen-chromatic ratio of a class of graph provides a form of energy distribution among the colour classes determined by the chromatic number of such a class of graphs. The asymptote associated with this eigen-chromatic ratio allows for the behavioural analysis in terms of stability of molecules in molecular graph theory where a large number of atoms are involved. This asymptote can be associated with the concept of graphs being hyper- or hypo- energetic (see [48])

The Victorian Newsletter (Fall 1981)

The Victorian Newsletter is sponsored for the Victorian Group of Modern Language Association by the Western Kentucky University and is published twice annually.Tennyson and Carlyle: A Source for "The Eagle" / Paul F. Mattheisen -- The Schooling of John Bull: Form and Moral in Talbot Baines Reed's Boys' Stories and in Kipling's Stalky & Co. / Patrick Scott -- How It Struck A Contemporary: Tennyson's "Lancelot and Elaine" and Pre-Raphaelite Art / Catherine Barnes Stevenson -- Amours de Voyage and Matthew Arnold in Love: An Inquiry / Eugene R. August -- Tennyson's "Ulysses" as Rhetorical Monologue / Mary Saunders -- The Mathematical References to the Adoption of the Gregorian Calendar in Lewis Carroll's Alice's Adventures in Wonderland / Laurence Dreyer -- Self-Helpers and Self-Seekers: Some Changing Attitudes to Wealth, 1840-1910 / J. L. Winte