Discrete Mathematics and Combinatorics Commons^™

Reducibility Of Eulerian Graphs And Digraphs, Akram B. Attar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper the concept of reducibility in graph theory is discussed, and the deletable vertex (edge) in graph (digraph) is defined. The class of graphs (digraphs) ℛ is called vertex (edge) reducible if for any G∈ ℛ either ℛ is the trivial graph (null graph) or it contains a vertex (edge) v such that G-v∈ ℛ. We introduce some classes of graphs (digraphs) which are reducible and others which are not. The vertex reducibility and edge reducibility of Eulerian graphs and Eulerian digraphs have also been studied.

Extremal Results For Peg Solitaire On Graphs, Aaron D. Gray

Electronic Theses and Dissertations

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families.

Application Of Linear Sequences To Cryptography, Amanda C. Yeates

Honors Theses

Cryptography is the study of a centuries–old technique of secretly transferring information between parties. Linear recurrences were the chosen method of encryption and decryption in the thesis. The Fibonacci sequence, with its Zeckendorf representation, allows for the flexibility of encoding any number desired based on a particular encoding technique used in the film Sherlock Holmes: A Game of Shadows. The main goal is to find other linear recurrences that possess characteristics similar to the Fibonacci sequence to use as suitable substitutes for encoding. Different sequences were analyzed based on a number of criteria. In order for a sequence to be …

Packings And Coverings Of Complete Graphs With A Hole With The 4-Cycle With A Pendant Edge, Yan Xia

Electronic Theses and Dissertations

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs K_v, (2) complete bipartite graphs K_m,n, and (3) complete graphs with a hole K(v,w).

Lattice Point Counting And Height Bounds Over Number Fields And Quaternion Algebras, Lenny Fukshansky, Glenn Henshaw

CMC Faculty Publications and Research

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number …

Counting The Number Of Squares Reachable In K Knight's Moves., Amanda M. Miller, David L. Farnsworth

Articles

from its initial position on an infinite chessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and 28k – 20 for k ≥ 5. The cumulative number of squares reach- able in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k-squared – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more mathematically accessible then preceding proofs.

A Discrete Approach To The Poincare-Miranda Theorem, Connor Thomas Ahlbach

HMC Senior Theses

The Poincare-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincare-Miranda Theorem that are discrete in nature - that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. …

Hypergraph Capacity With Applications To Matrix Multiplication, John Lee Thompson Peebles Jr.

HMC Senior Theses

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics.

We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such …

Chip Firing Games And Riemann-Roch Properties For Directed Graphs, Joshua Z. Gaslowitz

HMC Senior Theses

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

Very Cost Effective Partitions In Graphs, Inna Vasylieva

Electronic Theses and Dissertations

For a graph G=(V,E) and a set of vertices S, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V -S than in S.

A bipartition pi={S, V- S} is called very cost effective if both S and V- S are very cost effective sets. Not all graphs have a very cost effective bipartition, for example, the complete graphs of odd order do not. We consider several families of graphs G, including Cartesian products and cacti graphs, to determine whether G has a very cost effective bipartition.

Generalizations Of Pascal's Triangle: A Construction Based Approach, Michael Anton Kuhlmann

UNLV Theses, Dissertations, Professional Papers, and Capstones

The study of this paper is based on current generalizations of Pascal's Triangle, both the expansion of the polynomial of one variable and the multivariate case. Our goal is to establish relationships between these generalizations, and to use the properties of the generalizations to create a new type of generalization for the multivariate case that can be represented in the third dimension.

In the first part of this paper we look at Pascal's original Triangle with properties and classical applications. We then look at contemporary extensions of the triangle to coefficient arrays for polynomials of two forms. The first of …

Simulated Annealing Approach To Flow Shop Scheduling, Sadhana Yellanki

UNLV Theses, Dissertations, Professional Papers, and Capstones

Flow Shop Scheduling refers to the process of allotting various jobs to the machines given, such that every job starts to process on a machine n only after it has finished processing on machine n-1, with each job having n operations to be performed one per machine. To find a schedule that leads to the optimal utilization of resources, expects the schedule to finish in a minimum span of time, and also satisfy the optimality criterion set for the related scheduling problem is NP-Hard, if n > 2. In this thesis, we have developed an algorithm adopting a heuristic called Simulated …

Peg Solitaire On Trees With Diameter Four, Clayton A. Walvoort

Electronic Theses and Dissertations

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.

Restricted And Unrestricted Coverings Of Complete Bipartite Graphs With Hexagons, Wesley M. Surber

Electronic Theses and Dissertations

A minimal covering of a graph G with isomorphic copies of graph H is a set {H₁, H₂, H₃, ... , H_n} where H_i is isomorphic to H, the vertex set of H_i is a subset of G, the edge set of G is a subset of the union of H_i's, and the cardinality of the union of H_i's minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of H_{i …}

Information-Based Physics: An Intelligent Embedded Agent's Guide To The Universe, Kevin H. Knuth

Physics Faculty Scholarship

In this talk, I propose an approach to understanding the foundations of physics by considering the optimal inferences an intelligent agent can make about the universe in which he or she is embedded. Information acts to constrain an agent’s beliefs. However, at a fundamental level, any information is obtained from interactions where something influences something else. Given this, the laws of physics must be constrained by both the nature of such influences and the rules by which we can make inferences based on information about these influences. I will review the recent progress we have made in this direction. This …

The K-Dominating Graph, Ruth Haas, Karen Seyffarth

Mathematics Sciences: Faculty Publications

Abstract. Given a graph G, the k-dominating graph of G, D_k(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D_k(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D_k(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of …

Generalizing Tanisaki's Ideal Via Ideals Of Truncated Symmetric Functions, Aba Mbirika, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Abstract. We define a family of ideals I_h in the polynomial ring Z[_{x1, . . . , xn}] that are parametrized by Hessenberg functions h (equivalently Dyck paths or ample partitions). The ideals I_h generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define I_h, we use polynomials in a proper subset of the variables {_{x1, . . . , xn}} that are symmetric under the corresponding permutation subgroup. We call these polynomials truncated symmetric functions and …

Chromatic Bounds On Orbital Chromatic Roots, Dae Hyun Kim, Alexander H. Mun, Mohamed Omar

All HMC Faculty Publications and Research

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in R, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in [32/27,∞). Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root …

A New Recursion For Three-Column Combinatorial Macdonald Polynomials, Elizabeth Niese

Mathematics Faculty Research

The Hilbert series of the Garsia–Haiman module M_μ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n . Since there are n ! fillings that generate , it is desirable to find recursions to reduce the number of fillings that need to be considered when computing combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!ⁿn!).

Knight's Tours On 3 X N Chessboards With A Single Square Removed, Amanda M. Miller, David L. Farnsworth

Articles

The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or n = 11 when square (1, 3), (2, 4), …

Constructions And Enumeration Methods For Cubic Graphs And Trees, Erica R. Gilliland

Undergraduate Honors Thesis Collection

The goal of this thesis is to study two related problems that, in the broadest terms, lie in a branch of mathematics called graph theory. The first problem examines some new techniques for constructing a Hamilton graph of least possible order and having a preassigned girth, and the second concerns the enumeration of a certain type of graphs called trees.

Billey's Formula In Combinatorics, Geometry, And Topology, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of …

Set Theoretic Approach To Algebraic Structures In Mathematics - A Revelation, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out how sets in algebraic structure can be used to construct most generalized algebraic structures, like set linear algebra/vector space, set ideals in rings and semigroups. This sort of study is not only innovative but infact very helpful in cases instead of working with a large data we can work with a considerably small data. Thus instead of working with a vector space or a linear algebra V over a field F we can work with a subset in V and a needed subset in F, this can save both time and economy. The concept …

Subset Non Associative Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time we introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this specific topic is still rare. Authors have recently introduced the notion of subset algebraic structures. The maximum algebraic structure enjoyed by subsets with two binary operations is just a semifield and semiring, even if a ring or a field is used. In case semigroups or groups are used still the algebraic structure of the subset is only a semigroup. To construct a subset non associative …

Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. …

A Generalized White Noise Space Approach To Stochastic Integration For A Class Of Gaussian Stationary Increment Processes, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. Special inequalities which hold in this space allow to characterize its invertible elements and to develop an appropriate framework of non-commutative stochastic linear systems.

Topological Convolution Algebras, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce a new family of topological convolution algebras of the form ⋃p∈NL2(S,μp), where S is a Borel semi-group in a locally compact group G, which carries an inequality of the type ∥f∗g∥p≤Ap,q∥f∥q∥g∥p for p>q+d where d pre-assigned, and Ap,q is a constant. We give a sufficient condition on the measures μp for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.

On Discrete Analytic Functions: Products, Rational Functions, And Reproducing Kernels, Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.

Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, …