Thesis (M.S.)--Rochester Institute of Technology, 2006. / Typescript. Includes bibliographical references (leaves 63-72).

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 99-109). Also available in electronic version.

Thesis (Ph. D.)--West Virginia University, 2007. / Title from document title page. Document formatted into pages; contains vi, 49 p. : ill. Includes abstract. Includes bibliographical references (p. 45-49).

Chapter I begins with a brief history of the topic of greatest common subgraphs. Then we provide a summaiy of the work done on some variations of greatest common subgraphs. Finally, in this chapter we present results previously obtained on greatest common divisors and least common multiples of graphs. In Chapter II the concepts of prime graphs, prime divisors of graphs, and primeconnected graphs are presented. We show the existence of prime trees of any odd size and the existence of prime-connected trees that are not prime having any odd composite size. Then the number of prime divisors in a graph is studied. Finally, we present several results involving the existence of graphs whose size satisfies some prescribed condition and which contains a specified number of prime divisors. Chapter III presents properties of greatest common divisors and least common multiples of graphs. Then graphs with a prescribed number of greatest common divisors or least common multiples are studied. In Chapter IV we study the sizes of greatest common divisors and least common multiples of specified graphs. We find the sizes of greatest common divisors and least common multiples of stars and that of stripes. Then the size of greatest common divisors and least common multiples of paths and complete graphs are investigated. In particular, the size of least common multiples of paths versus K3 or K4 are determined. Then we present the greatest common divisor index of a graph and we determine this parameter for several classes of graphs. iii In Chapter V greatest common divisors and least common multiples of digraphs are introduced. The existence of least common mutliples of two stars is established, and the size of a least common multiple is found for several pairs of stars. Finally, we present the concept of greatest common divisor index of a digraph and determine it for several classes of digraphs. iv / Mathematical Sciences / Ph. D. (Mathematical sciences)

Let μ be an eigenvalue of the graph G with multiplicity k. A star set corresponding to μ in G is a subset of V(G) such that [x] = k and μ is not an eigenvalue of G - X. It is always the case that the vertex set of G can be partitioned into star sets corresponding to the distinct eigenvalues of G. Such a partition is called a star partition. We give some examples of star partitions and investigate the dominating properties of the set V (G) \ X when μ ε {-I, a}. The induced subgraph H = G - X is called a star complement for μ in G. The Reconstruction Theorem states that for a given eigenvalue μ of G, knowledge of a star complement corresponding to μ, together with knowledge of the edge set between X and its complement X, is sufficient to reconstruct G. Pursuant to this we explore the idea that the adjacencies of pairs of vertices in X is determined by the relationship between the H-neighbourhoods of these vertices. We give some new examples of cubic graphs in this context. For a given star complement H the range of possible values for the corresponding eigenvalue μ is constrained by the condition that μ must be a simple eigenvalue of some one-vertex extension of H, and a double eigenvalue of some two-vertex extension of H. We apply the Reconstruction Theorem to the generic form of a two-vertex extension of H, thereby obtaining sufficient information to construct a graph containing H as a star complement for one of the possible eigenvalues. We give examples of graph characterizations arising in the case where the star complement is (to within isolated vertices) a complete bipartite graph.

Ph.D. (Mathematics) / Please refer to full text to view abstract

M.Sc. (Mathematics) / In chapter 1, we give the reader some background concerning digraphs that are used in the discussion of Markov chains; namely, their Markov digraphs. Warshall’s Algorithm for reachability is also introduced as this is used to define terms such as transient states and irreducibility. Some initial theory and definitions concerning Markov chains and their corresponding Markov digraphs are given in chapter 2. Furthermore, we discuss l–step transitions (walks of length l) for homogeneous and inhomogeneous Markov chains and other generalizations. In chapter 3, we define terms such as communication, intercommunication, recurrence and transience. We also prove some results regarding the irreducibility of some Markov chains through the use of the reachability matrix. Furthermore, periodicity and aperiodicity are also investigated and the existence of walks of any length greater than some specified integer N is also considered. A discussion on random walks on an undirected torus is also contained in this chapter. In chapter 4, we explore stationary distributions and what it means for a Markov chain to be reversible. Furthermore, the hitting time and the mean hitting time in a Markov digraph are also defined and the proof of the theorems regarding them are done. The demonstrations of the theorems concerning the existence and uniqueness of stationary distributions and the Markov Chain Convergence Theorem are carried out. Later in this chapter, we define the Markov digraph of undirected graphs, which are Markov chains as well. The existing theory is then applied to these. In chapter 5, we explore and see how to simulate Markov chains on a computer by using Markov Chain Monte Carlo Methods. We also show how these apply to random q–colourings of undirected graphs. Finally, in chapter 6, we consider a physical application of these Graph Theoretical concepts—the simulation of the Ising model. Initially, the relevant concepts of the Potts model are given and then the Gibbs sampler algorithm in chapter 5 is modified and used to simulate the Ising model. A relation between the chromatic polynomial and the partition function of the Potts model is also demonstrated.

D.Phil. (Mathematics) / Domination and its variants are now well studied in graph theory. One of these variants, paired-domination, requires that the subgraph induced by the dominating set contains a perfect matching. In this thesis, we further investigate the concept of paired-domination. Chapters 2, 3, 4, and 5 of this thesis have been published in [17], [41], [42], and [43], respectively, while Chapter 6 is under submission; see [44]. In Chapter 1, we introduce the domination parameters we use, as well as the necessary graph theory terminology and notation. We combine the de nition of a paired-dominating set and a locating set to de ne three new sets: locating-paired- dominating sets, di erentiating-paired-dominating sets, and metric-locating-paired- dominating sets. We use these sets in Chapters 3 and 4. In Chapter 2, we investigate the relationship between the upper paired-domination and upper total domination numbers of a graph. In Chapter 3, we study the properties of the three kinds of locating paired-dominating sets we de ned, and in Chapter 4 we give a constructive characterisation of those trees which do not have a di erentiating- paired-dominating set. In Chapter 5, we study the problem of characterising planar graphs with diameter two and paired-domination number four. Lastly, in Chap- ter 6, we establish an upper bound on the size of a graph of given order and paired- domination number and we characterise the extremal graphs that achieve equality in the established bound.

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