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Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius
Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius
Graduate College Dissertations and Theses
In this work, the current state of the field of edge-colored graphs is surveyed. Anew concept of unshrinkable edge colorings is introduced which is useful for rainbow subgraph problems and interesting in its own right. This concept is analyzed in some depth. Building upon the linear edge ordering described in a recent work from Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer, edge-ordering graphs with the cyclic group is introduced and some results are given on this and a related counting problem.
Strategies Community College Mexican American Adult College Algebra Students Use When Graphing Function Transformations, Roxana Pamela Jimenez
Strategies Community College Mexican American Adult College Algebra Students Use When Graphing Function Transformations, Roxana Pamela Jimenez
Theses and Dissertations
This qualitative case study pursued to describe the different strategies Mexican American adult students in a local community college used to graph function transformations. Participants in the study were purposefully selected using a criterion sampling to ensure participants were atypical, above average students between the ages 18-22, and had a final course average of 89.5-100 in College Algebra. Three research questions were examined 1) In what ways do Mexican American adult college students graph a function transformation? 2) Which strategies do students implement when graphing a function transformation? Qualitative research methods using think aloud semi-structured interviews were used in this …
Cohen-Macaulay Properties Of Closed Neighborhood Ideals, Jackson Leaman
Cohen-Macaulay Properties Of Closed Neighborhood Ideals, Jackson Leaman
All Theses
This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in …
Matroid Generalizations Of Some Graph Results, Cameron Crenshaw
Matroid Generalizations Of Some Graph Results, Cameron Crenshaw
LSU Doctoral Dissertations
The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.
For a graph G, the ratio of |E(G)| to the minimum degree of G has a natural lower bound. For a matroid M that is representable over a finite field, we generalize this to a lower bound on …
Opposite Trees, Theo Goossens
Opposite Trees, Theo Goossens
Theses and Dissertations (Comprehensive)
A spanning tree of a graph G is a connected acyclic subgraph of G that includes all of the vertices in G. The degree of a vertex is the number of edges incident to that vertex. Given a spanning tree T of a graph G, an opposite tree of T is a spanning tree of G where the degree of each of its vertices is different from its degree in T. For complete, complete bipartite, and complete multipartite graphs, we give the conditions spanning trees of these graphs must satisfy in order to have an opposite tree.
Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz
Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz
Honors College Theses
The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …
Proper Sum Graphs, Austin Nicholas Beard
Proper Sum Graphs, Austin Nicholas Beard
MSU Graduate Theses
The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.
On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece
On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece
MSU Graduate Theses
In this paper we discuss the Hamiltonicity of the subgroup lattices of
different classes of groups. We provide sufficient conditions for the
Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also
prove the non-Hamiltonicity of the subgroup lattices of dihedral and
dicyclic groups. We disprove a conjecture on non-abelian p-groups by
producing an infinite family of non-abelian p-groups with Hamiltonian
subgroup lattices. Finally, we provide a list of the Hamiltonicity of the
subgroup lattices of every finite group up to order 35 barring two groups.
Jacobians Of Finite And Infinite Voltage Covers Of Graphs, Sophia Rose Gonet
Jacobians Of Finite And Infinite Voltage Covers Of Graphs, Sophia Rose Gonet
Graduate College Dissertations and Theses
The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a graph X; it is a finite abelian group whose cardinality is equal to the number of spanning trees of X (Kirchhoff's Matrix Tree Theorem). This dissertation proves results about the Jacobians of certain families of covering graphs, Y, of a base graph X, that are constructed from an assignment of elements from a group G to the edges of X (G is called the voltage group and Y is called the derived graph). The principal aim is to relate the Jacobian of …
The Game Of Cops And Robbers On Planar Graphs, Jordon S. Daugherty
The Game Of Cops And Robbers On Planar Graphs, Jordon S. Daugherty
MSU Graduate Theses
In graph theory, the game of cops and robbers is played on a finite, connected graph. The players take turns moving along edges as the cops try to capture the robber and the robber tries to evade capture forever. This game has received quite a bit of recent attention including several conjectures that have yet to be proven. In this paper, we restricted our attention to planar graphs in order to try to prove the conjecture that the dodecahedron graph is the smallest planar graph, in terms of vertices, that has cop number three. Along the way we discuss several …
Demystification Of Graph And Information Entropy, Bryce Frederickson
Demystification Of Graph And Information Entropy, Bryce Frederickson
Undergraduate Honors Capstone Projects
Shannon entropy is an information-theoretic measure of unpredictability in probabilistic models. Recently, it has been used to form a tool, called the von Neumann entropy, to study quantum mechanics and network flows by appealing to algebraic properties of graph matrices. But still, little is known about what the von Neumann entropy says about the combinatorial structure of the graphs themselves. This paper gives a new formulation of the von Neumann entropy that describes it as a rate at which random movement settles down in a graph. At the same time, this new perspective gives rise to a generalization of von …
A Mathematical Analysis Of The Game Of Santorini, Carson Clyde Geissler
A Mathematical Analysis Of The Game Of Santorini, Carson Clyde Geissler
Senior Independent Study Theses
Santorini is a two player combinatorial board game. Santorini bears resemblance to the graph theory game of Geography, a game of moving and deleting vertices on a graph. We explore Santorini with game theory, complexity theory, and artificial intelligence. We present David Lichtenstein’s proof that Geography is PSPACE-hard and adapt the proof for generalized forms of Santorini. Last, we discuss the development of an AI built for a software implementation of Santorini and present a number of improvements to that AI.
On Fractional Realizations Of Tournament Score Sequences, Kaitlin S. Murphy
On Fractional Realizations Of Tournament Score Sequences, Kaitlin S. Murphy
All Graduate Theses and Dissertations, Spring 1920 to Summer 2023
Contrary to popular belief, we can’t all be winners.
Suppose 6 people compete in a chess tournament in which all pairs of players compete directly and no ties are allowed; i.e., 6 people compete in a ‘round robin tournament’. Each player is assigned a ‘score’, namely the number of games they won, and the ‘score sequence’ of the tournament is a list of the players’ scores. Determining whether a given potential score sequence actually is a score sequence proves to be difficult. For instance, (0, 0, 3, 3, 3, 6) is not feasible because two players cannot both have …
Unavoidable Immersions And Intertwines Of Graphs, Matthew Christopher Barnes
Unavoidable Immersions And Intertwines Of Graphs, Matthew Christopher Barnes
LSU Doctoral Dissertations
The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with …
On Some Geometry Of Graphs, Zachary S. Mcguirk
On Some Geometry Of Graphs, Zachary S. Mcguirk
Dissertations, Theses, and Capstone Projects
In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner's inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size of …
Exploring Sequences Of Tournament Graphs With Draws, Kaylynn H. Tran
Exploring Sequences Of Tournament Graphs With Draws, Kaylynn H. Tran
Senior Projects Spring 2018
Tournaments occur all over the world and they are used to decide championships in various competitions. For this project, I will be exploring tournaments in the round robin style in which every team plays one another. This is based on Sadiki Lewis' senior project, Exploring Tournament Graphs and Their Win Sequences. I will be expanding his project by including the possibility of a draw between two teams, in addition to a win or a loss. Teams and games will be modeled by complete graphs where each vertex represents a team and each directed edge between two vertices represents the outcome …
Graph Homomorphisms And Vector Colorings, Michael Robert Levet
Graph Homomorphisms And Vector Colorings, Michael Robert Levet
Theses and Dissertations
A graph vertex coloring is an assignment of labels, which are referred to as colors, such that no two adjacent vertices receive the same color. The vertex coloring problem is NP-Complete [8], and so no polynomial time algorithm is believed to exist. The notion of a graph vector coloring was introduced as an efficiently computable relaxation to the graph vertex coloring problem [7]. In [6], the authors examined the highly symmetric class of 1-walk regular graphs, characterizing when such graphs admit unique vector colorings. We present this characterization, as well as several important consequences discussed in [5, 6]. By appealing …
Stationary Automata, Anaam Mutar Bidhan
Stationary Automata, Anaam Mutar Bidhan
Graduate Theses, Dissertations, and Problem Reports
In this dissertation, we investigate new automata, we call it stationary automata or ST-automata. This concept is based on the definition of TF-automaton by Wojciechowski [Woj2]. What is new in our approach is that we incorporate stationary subsets of limit ordinals of uncountable cofinality. The first objective of the thesis is to motivate the new construction of automata. This concept of ST-automata allows us to make a connection with infinite graph theory. Aharoni, Nash-Williams, and Shelah [AhNaSh] formulated a condition that is necessary and sufficient for a bipartite graph to have a matching. For a bipartite …
Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore
Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log2(α)3 + 16.5log2(α)2 + …
Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber
Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber
Doctoral Dissertations
In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.
An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring R and ideal I the vertices are { …
The Element Spectrum Of A Graph, Milisha Hart-Simmons
The Element Spectrum Of A Graph, Milisha Hart-Simmons
Electronic Theses and Dissertations
Characterizations of graphs and matroids that have cycles or circuits of specified cardinality have been given by authors including Edmonds, Junior, Lemos, Murty, Reid, Young, and Wu. In particular, a matroid with circuits of a single cardinality is called a Matroid Design. We consider a generalization of this problem by assigning a weight function to the edges of a graph. We characterize when it is possible to assign a positive integer value weight function to a simple 3-connected graph G such that the graph G contains an edge that is only in cycles of two different weights. For example, as …
Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri
Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri
HMC Senior Theses
Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through …
A Survey Of Graphs Of Minimum Order With Given Automorphism Group, Jessica Alyse Woodruff
A Survey Of Graphs Of Minimum Order With Given Automorphism Group, Jessica Alyse Woodruff
Math Theses
We survey vertex minimal graphs with prescribed automorphism group. Whenever possible, we also investigate the construction of such minimal graphs, confirm minimality, and prove a given graph has the correct automorphism group.
P_4-Decomposability In Regular Graphs And Multigraphs, David Joshua Mendell
P_4-Decomposability In Regular Graphs And Multigraphs, David Joshua Mendell
Theses and Dissertations
The main objective of this thesis is to review and expand the study of graph decomposability. An H-decomposition of a graph G=(V,E) is a partitioning of the edge set, $E$, into edge-disjoint isomorphic copies of a subgraph H. In particular we focus on the decompositions of graphs into paths. We prove that a 2,4 mutligraph with maximum multiplicity 2 admits a C_2,C_3-free Euler tour (and thus, a decomposition into paths of length 3 if it has size a multiple of 3) if and only if it avoids a set of 15 forbidden structures. We also prove that …
Properties Of Ideal-Based Zero-Divisor Graphs Of Commutative Rings, Jesse Gerald Smith Jr.
Properties Of Ideal-Based Zero-Divisor Graphs Of Commutative Rings, Jesse Gerald Smith Jr.
Doctoral Dissertations
Let R be a commutative ring with nonzero identity and I an ideal of R. The focus of this research is on a generalization of the zero-divisor graph called the ideal-based zero-divisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zero-divisor graph of R. We begin by classifying all rings which have nontrivial ideal-based zero-divisor graph complete on fewer than 5 vertices. We also classify when such graphs are complete on a prime number of vertices. In addition we classify all rings which …
The Minimum Rank Of Schemes On Graphs, William Nelson Sexton
The Minimum Rank Of Schemes On Graphs, William Nelson Sexton
Theses and Dissertations
Let G be an undirected graph on n vertices and let S(G) be the class of all real-valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let V = {1, 2, . . . , n} be the vertex set of G. A scheme on G is a function f : V → {0, 1}. Given a scheme f on G, there is an associated class of matrices Sf (G) = {A ∈ S(G)|aii = 0 if and only if f(i) = 0}. A scheme f is said …
Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis
Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis
Masters Theses
We examine properties of graphs that result in the graph having a contractible theta complex. We classify such properties for tree graphs and graphs with one loop and we introduce examples of graphs with such properties for tree graphs and graphs with one or two loops. For more general graphs, we show that having a contractible theta complex is not an elusive property, and that any skeleton of a graph with at least three loops can be made to have a contractible theta complex by strategically adding vertices to its skeleton.
Diagonal Entry Restrictions In Minimum Rank Matrices, And The Inverse Inertia And Eigenvalue Problems For Graphs, Curtis G. Nelson
Diagonal Entry Restrictions In Minimum Rank Matrices, And The Inverse Inertia And Eigenvalue Problems For Graphs, Curtis G. Nelson
Theses and Dissertations
Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in …
Independence Polynomials Of Molecular Graphs, Cameron Taylor Byrum
Independence Polynomials Of Molecular Graphs, Cameron Taylor Byrum
Electronic Theses and Dissertations
In the 1980's, it was noticed by molecular chemists that the stability and boiling point of certain molecules were related to the number of independent vertex sets in the molecular graphs of those chemicals. This led to the definition of the Merrifield-Simmons index of a graph G as the number of independent vertex sets in G. This parameter was extended by graph theorists, who counted independent sets of different sizes and defined the independence polynomial F_G(x) of a graph G to be \sum_k F_k(G)x^k where for each k, F_k(G) is the number of independent sets of k vertices. This thesis …
The Minimum Rank, Inverse Inertia, And Inverse Eigenvalue Problems For Graphs, Mark Condie Kempton
The Minimum Rank, Inverse Inertia, And Inverse Eigenvalue Problems For Graphs, Mark Condie Kempton
Theses and Dissertations
For a graph G we define S(G) to be the set of all real symmetric n by n matrices whose off-diagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.