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Articles 1 - 10 of 10
Full-Text Articles in Algebra
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 …
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.
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 { …
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.
Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral
Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …
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 …