Physical Sciences and Mathematics Commons^™

Bounds For The Independence Number Of A Graph, William Willis

Theses and Dissertations

The independence number of a graph is the maximum number of vertices from the vertex set of the graph such that no two vertices are adjacent. We systematically examine a collection of upper bounds for the independence number to determine graphs for which each upper bound is better than any other upper bound considered. A similar investigation follows for lower bounds. In several instances a graph cannot be found. We also include graphs for which no bound equals $\alpha$ and bounds which do not apply to general graphs.

Transcriptomic Data Analysis Using Graph-Based Out-Of-Core Methods, Gary L Rogers

Doctoral Dissertations

Biological data derived from high-throughput microarrays can be transformed into finite, simple, undirected graphs and analyzed using tools first introduced by the Langston Lab at the University of Tennessee. Transforming raw data can be broken down into three main tasks: data normalization, generation of similarity metrics, and threshold selection. The choice of methods used in each of these steps effect the final outcome of the graph, with respect to size, density, and structure. A number of different algorithms are examined and analyzed to illustrate the magnitude of the effects.

Graph-based tools are then used to extract putative gene networks. These …

A Characterization Of Ramsey Graphs For R(3,4), Nicholas M. Richardson

Doctoral Dissertations

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to …

A Characterization Of Ramsey Graphs For R(3,4), Nicholas M. Richardson

Doctoral Dissertations

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to …

Parity Domination In Product Graphs, Christopher Whisenant

Theses and Dissertations

An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating …

A Predictive Model Which Uses Descriptors Of Rna Secondary Structures Derived From Graph Theory., Alissa Ann Rockney

Electronic Theses and Dissertations

The secondary structures of ribonucleic acid (RNA) have been successfully modeled with graph-theoretic structures. Often, simple graphs are used to represent secondary RNA structures; however, in this research, a multigraph representation of RNA is used, in which vertices represent stems and edges represent the internal motifs. Any type of RNA secondary structure may be represented by a graph in this manner. We define novel graphical invariants to quantify the multigraphs and obtain characteristic descriptors of the secondary structures. These descriptors are used to train an artificial neural network (ANN) to recognize the characteristics of secondary RNA structure. Using the ANN, …

Universal Hypergraphs., Michael Deren

Electronic Theses and Dissertations

In this thesis, we study universal hypergraphs. What are these? Let us start with defining a universal graph as a graph on n vertices that contains each of the many possible graphs of a smaller size k < n as an induced subgraph. A hypergraph is a discrete structure on n vertices in which edges can be of any size, unlike graphs, where the edge size is always two. If all edges are of size three, then the hypergraph is said to be 3-uniform. If a 3-uniform hypergraph can have edges colored one of a colors, then it is called a …

The Roller-Coaster Conjecture, Leslie A. Cheteyan

Theses, Dissertations and Culminating Projects

No abstract provided.

Zero-Sum Magic Graphs And Their Null Sets, Samuel M. Hansen

UNLV Theses, Dissertations, Professional Papers, and Capstones

For any element h of the Natural numbers, a graph G=(V,E), with vertex set V and edge set E, is said to be h-magic if there exists a labeling of the edge set E, using the integer group mod h such that the induced vertex labeling, the sum of all edges incident to a vertex, is a constant map. When this constant is 0 we call G a zero-sum h-magic graph. The null set of G is the set of all natural numbers h for which G admits a zero-sum h-magic labeling. A graph G is said to be uniformly …

Embeddings Of Product Graphs Where One Factor Is A Hypercube, Bethany Turner

Theses and Dissertations

Voltage graph theory can be used to describe embeddings of product graphs if one factor is a Cayley graph. We use voltage graphs to explore embeddings of various products where one factor is a hypercube, describing some minimal and symmetrical embeddings. We then define a graph product, the weak symmetric difference, and illustrate a voltage graph construction useful for obtaining an embedding of the weak symmetric difference of an arbitrary graph with a hypercube.

Results In Lattices, Ortholattices, And Graphs, Jianning Su

Doctoral Dissertations

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "π-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider …

Hückel Energy Of A Graph: Its Evolution From Quantum Chemistry To Mathematics, Steven Zimmerman

Electronic Theses and Dissertations

The energy of a graph began with German physicist, Erich H¨uckel’s 1931 paper, Quantenttheoretische Beitr¨age zum Benzolproblem. His work developed a method for computing the binding energy of the π-electrons for a certain class of organic molecules. The vertices of the graph represented the carbon atoms while the single edge between each pair of distinct vertices represented the hydrogen bonds between the carbon atoms. In turn, the chemical graphs were represented by an n × n matrix used in solving Schr¨odinger’s eigenvalue/eigenvector equation. The sum of the absolute values of these graph eigenvalues represented the total π-electron energy. The criteria …

Finding Dud Vertices In Defensive Alliances And Secure Sets Using Computational Tools, George Worley Ii

Electronic Theses and Dissertations

Defensive alliances are a way of using graphs to model the defense of resources (people, buildings, countries, etc.) against attacks where the number of potential attackers against each resource is known. The initial study of defensive alliances focused on questions of minimal defensive alliances in a graph and the minimum possible size of a defensive alliance in a graph, but in order to apply defensive alliances in modeling real-world situations, additional considerations are important. In particular, since each vertex in a defensive alliance represents some real-world object that has a cost associated with remaining in the defensive alliance, it is …