D.Phil.

M.Sc. (Mathematics) / Please refer to full text to view abstract.

Philosophiae Doctor - PhD / In this thesis only Binary codes are studied. Firstly, the codes overs the field GF(2) by the adjacency matrix of the complement T(n), ofthe triangular graph, are examined. It is shown that the code obtained is the full space F2 s(n/2) when n= 0 (mod 4) and the dual code of the space generated by the j-vector when n= 2(mod 4). The codes from the other two cases are less trivial: when n=1 (mod 4) the code is [(n 2), (n 2 ) - n + 1, 3] code, and when n = 3 (mod 4) it is an [(n 2), (n 2) - n, 4 ] code. / South Africa

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The purpose of this thesis is to investigate the graph isomorphism problem for a special class of graphs. Each graph is characterized by its edge set, and a subgroup of its automorphism group, called the colour group. In particular, a simple, efficient algorithm for determining whether two graphs are isomorphic if at least one is a member of the class is developed. Chapter 1 provides some basic definitions and lemmas required in the text. The concepts of reducibility and reducible bipartite graphs are introduced, and the properties of the colour groups of such graphs are investigated. Chapter 2 establishes some results concerning the existence of reducible graphs. Conditions based on the existence of vertices with prescribed properties are shown to provide sufficient conditions for a graph to be reducible. In the special case of trees they are shown to be both necessary and sufficient. Necessary conditions for the reducibility of graphs, based on their radius and diameter are also established. Chapter 3 describes an algorithm for determining whether a graph is completely reducible, which is applied to a test for isomorphism. An investigation of the speed of this algorithm is made and its efficiency is compared with an algorithm of D. Corneil [5], which this author considers the best for arbitrary graphs in the current literature. / Science, Faculty of / Computer Science, Department of / Graduate

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Let $l$ be a graceful label of a graceful graph $G$ with $n$ nodes. We outline a procedure to generate a graceful label $l$ and its graph $G$ by constructing a sequence of labeled edges $(a_k)_{k=1}^{n-1}$ where the $k$th term of the sequence corresponds to an edge labeled $k$. We use the complement of the label generated to identify a class of transformations on graceful labels that can produce additional graceful labelings on $G$. We then identify a subset of labels generated this way with properties that limit the number of graceful labels such a graph can have and study some properties of those labels. We prove that all edge-preserving transformations of these labels fix over half of all node labels, and after establishing criteria necessary for such a transformation to leave some node labels unfixed, we show that for $n\leq9$ these transformations fix all node labels.

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The well studied class of irredundant vertex sets of a graph has been previously shown to be a special case of (a) a “Private Neighbor Cube” of eight classes of vertex subsets and (b) a family of sixty four classes of “generalised irredundant sets.” The thesis makes various advances in the theory of irredundance. More specifically: (i) Nordhaus-Gaddum results for all the sixty-four classes of generalised irredundant sets are obtained. (ii) Sharp lower bounds involving order and maximum degree are attained for two specific classes in the Private Neighbor Cube. (iii) A new framework which includes both of the above generalisations and various concepts of domination, is proposed. / Graduate

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