Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; 1 v. (various pagings) : ill. (some col.). Vita. Includes abstract. Includes bibliographical references.

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 41 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 40-41).

Given a graph G=(V,E), a vertex ordering of G is a total order v1,v2,...,vn of V. A graph search algorithm is a systematic method for visiting each vertex in a graph, naturally producing a vertex ordering of the graph. We explore the problem of determining whether a given vertex in a graph can be the end (last) vertex of a search ordering for various common graph search algorithms when restricted to various graph classes, as well as the related problem of determining if a vertex is an end-vertex when a start vertex is specified for the search. The former is referred to as the end-vertex problem, and the latter is the beginning-end-vertex problem. For the beginning-end-vertex problem, we show it is NP-complete on bipartite graphs as well as degree restricted bipartite graphs for Lexicographic Breadth First Search, but solvable in polynomial time on split graphs for Breadth First Search. We show that the end-vertex problem is tractable for Lexicographic Breadth First Search on proper interval bigraphs and for Lexicographic Depth First Search on chordal graphs. Further, we show that the problem is NP-complete for Lexicographic Breadth First Search and Depth First Search on bipartite graphs. / Graduate

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j, k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j, k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)-mixed graphs) and k−edge-coloured graphs ((0, k)−mixed graphs). A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. An m−colouring of a (j, k)−mixed graph is a homomorphism from that graph to a target with m vertices. The (j, k)−chromatic number of a (j, k)−mixed graph is the least m such that an m−colouring exists. When (j, k) = (0, 1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring. Similarly, when (j, k) = (1, 0) and (j, k) = (0, k) these definitions are consistent with the usual definitions of homomorphism and colouring for oriented graphs and k−edge-coloured graphs, respectively. In this thesis we study the (j, k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1, 0)−chromatic number, more commonly called the oriented chromatic number, and the (0, k)−chromatic number. In examining oriented graphs, we provide improvements to the upper and lower bounds for the oriented chromatic number of the families of oriented graphs with maximum degree 3 and 4. We generalise the work of Sherk and MacGillivray on the 2−dipath chromatic number, to consider colourings in which vertices at the ends of iii a directed path of length at most k must receive different colours. We examine the implications of the work of Smolikova on simple colourings to study of the oriented chromatic number of the family of oriented planar graphs. In examining k−edge-coloured graphs we provide improvements to the upper and lower bounds for the family of 2−edge-coloured graphs with maximum degree 3. In doing so, we define the alternating 2−path chromatic number of k−edge-coloured graphs, a parameter similar in spirit to the 2−dipath chromatic number for oriented graphs. We also consider a notion of simple colouring for k−edge-coloured graphs, and show that the methods employed by Smolikova ́ for simple colourings of oriented graphs may be adapted to k−edge-coloured graphs. In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs. / Graduate

published_or_final_version / abstract / toc / Computer Science and Information Systems / Doctoral / Doctor of Philosophy

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In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

Given two set systems $\mathscr{A}$ and $\mathscr{B}$ over an $n$-element set, we say that $(\mathscr{A,B})$ forms a recovering pair if the following conditions hold: \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr{B}$, $A \setminus B = A' \setminus B' \Rightarrow A=A'$ \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr {B}$, $B \setminus A = B' \setminus A' \Rightarrow B=B'$ \\ In 1989, G\'bor Simonyi conjectured that if $(\mathscr)$ forms a recovering pair, then $|\mathscr||\mathscr|\leq 2^n$. This conjecture is the focus of this thesis. This thesis contains a collection of proofs of special cases that together form a complete proof that the conjecture holds for all values of $n$ up to 8. Many of these special cases also verify the conjecture for certain recovering pairs when $n>8$. We also present a result describing the nature of the set of numbers over which the conjecture in fact holds. Lastly, we present a new problem in graph theory, and discuss a few cases of this problem.

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept is related to the energy of a class of molecules in chemistry and was first brought to mathematics by Gutman in 1978 ([8]). In this thesis, we do a comprehensive study on the energy of graphs and digraphs. In Chapter 3, we review some existing upper and lower bounds for the energy of a graph. We come up with some new results in this chapter. A graph with n vertices is hyper-energetic if its energy is greater than 2n−2. Some classes of graphs are proved to be hyper-energetic. We find a new class of hyper-energetic graphs which is introduced and proved to be hyper-energetic in Section 3.3. The energy of a digraph is the sum of the absolute values of the real part of the eigenvalues of its adjacency matrix. In Chapter 4, we study the energy of digraphs in a way that Pe˜na and Rada in [19] have defined. Some known upper and lower bounds for the energy of digraphs are reviewed. In Section 4.5, we bring examples of some classes of digraphs in which we find their energy. Keywords. Energy of a graph, hyper-energetic graph, energy of a digraph. / vii, 80 leaves ; 29 cm