Discrete Mathematics and Combinatorics Commons^™

Finding Combinatorial Patterns In Real Valued Omics Data, Kenneth Smith

Dissertations

Precision medicine is a healthcare approach which tailors disease prevention and treatment to an individual, based on their genetics, environment, lifestyle, and physiological state. These factors interact to produce biological changes that can be measured to produce data called omics, and include genomics, lipidomics, and proteomics. Despite the abundance of omics data and analysis techniques, researchers still struggle to identify biological findings that replicate across data sets and translate into clinical applications. In this dissertation, we employ combinatorial optimization techniques to improve upon three steps in the precision medicine analysis pipeline: 1) data cleaning, 2) community detection, and 3) feature …

The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

LSU Doctoral Dissertations

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …

2-Accessibility Of The Lucas Numbers/Fibonacci Like Sequences/Wythoff Array, Cameron Lejeune

Mathematics Senior Capstone Papers

In this project we will be delving into the combinatorics side of mathematics, with basic a graph theory idea such as set coloring. It will be a continuation of a question formed by the works of Bruce M. Landman, and Aaron Robertson, ”3-Accessibility of the Fibonacci Numbers.” They were able to prove the Fibonacci Numbers to be 2-Accessible, and while they did not supply the induction proof, we were able to construct and provide a proof. This project will use their lemmas and propositions to attempt to prove 2-Accessibility of the Lucas Numbers, Fibonacci-like sequences, and the Zeckendorf-Wythoff Array.

An Exploration Of The Sums Of Two Squares And Pentagonal Numbers, Jacob Von Tress

Mathematics Senior Capstone Papers

In the field of number theory, square numbers are very significant, and finding the sums of square numbers is a topic of certain interest to mathematicians. The most immediate application for adding together two square numbers is to identify Pythagorean triples. However, apart from seeking sums of two squares that are squares themselves, interesting patterns emerge that have fascinated number theorists for decades. Particularly, the distribution of a number’s divisors can explicitly determine how many ways that number can be written as a sum of two squares. Furthermore, pentagonal numbers, similar to square numbers, can be visualized by drawing a …

Matroids Stemming From The Maximal Relaxation Of Graphic Matroids, Landen Nguyen

Mathematics Senior Capstone Papers

This research explores the base perspective for maximal relaxation of graphic matroids. We begin by covering the required knowledge of graph theory and the basics of matroid theory required to conduct this research such as definitions of matroids up to the definitions of relaxation, hyperplanes, and uniform matroids. We then analyze the conjecture in matroids obtained from simple connected graphs.

Four Colorful Methods For Finding The Chromatic Polynomial, Emerson Statom

Mathematics Senior Capstone Papers

Mathematicians apply algebraic graph theory to address and interpret problems arising out of data structures and optimization. For example, graph coloring problems have intrigued mathematicians and computer scientists alike. The chromatic polynomial was created to help solve such problems. This research aims to create a greater understanding of the underlying structures behind chromatic polynomials by analyzing the algebraic, deletion-contraction, and color partition methods. Each method has differing applications, but by viewing them all together, this research hopes to show the depth of this most interesting subject in a condensed manner.

Congruences Of The Sums Of The First Rp Fibonacci Numbers, Tyler Warzynak

Mathematics Senior Capstone Papers

This paper serves as an extension/application of a method detailed by Chang et. al. in two separate papers which worked with sums modulo p for combinatorial sequences, positive integers r, and prime numbers p. Using a known formula, the Fibonacci numbers were converted into a sum of binomial coefficients to apply the aforementioned method. A hypothesis was formed using computational methods, where a pattern was observed to hold for the first 50,000 prime numbers. This claim was then proven following the methodology from Chang et. al. using properties of Laurent polynomials, definitions and theorems related to the Fibonacci numbers, and …

Discrete Macaulay-Steiner Geometry, Nikola Kuzmanovski

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

This thesis is concerned with discrete isoperimetric inequalities and Hilbert functions. Two generalizations of the Ahlswede-Cai local global principle are presented. These results give positive answers to two questions posed by Harper. One of these results is achieved by proving uniqueness of the lexicographic and colexicographic orders in two dimensions. The other result generalizes the technique which is commonly known as compression and includes almost all previously published results in this direction. The Ahlswede-Cai local global principle is a direct corollary of this result. Optimal downsets are studied in rectangles and triangles. All optimal downsets are found. The main result …

On Generating Bijections For Permutations And Inversion Sequences, Melanie J. Ferreri

Dartmouth College Ph.D Dissertations

Given an algebraic proof of a combinatorial identity, we use recursive methods to construct a bijection demonstrating the identity.

Our first application centers around derangements and nonderangements. A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of n elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this extends …

Wang Tilings In Arbitrary Dimensions, Ian Tassin

Rose-Hulman Undergraduate Mathematics Journal

This paper makes a new observation about arbitrary dimensional Wang Tilings,
demonstrating that any d -dimensional tile set that can tile periodically along d − 1 axes must be able to tile periodically along all axes.
This work also summarizes work on Wang Tiles up to the present day, including
definitions for various aspects of Wang Tilings such as periodicity and the validity of a tiling. Additionally, we extend the familiar 2D definitions for Wang Tiles and associated properties into arbitrary dimensional spaces. While there has been previous discussion of arbitrary dimensional Wang Tiles in other works, it has been …

Strongly I-Bicritical Graphs, Michelle Edwards, Gary Macgillivray, Shahla Nasserasr

Theory & Applications of Graphs

A graph $G$ is \emph{strongly $i$-bicritical} if it has independent domination number $i(G) \geq 3$, and $i(G - \{x, y\}) = i(G) - 2$ whenever $x$ and $y$ are two non-adjacent vertices of $G$. We describe five constructions of strongly $i$-bicritical graphs. For four of them, necessary and sufficient conditions for the graph produced by the construction to be strongly $i$-bicritical are given. The strongly $i$-bicritical graphs with independent domination number $i(G) = 3$ are characterized, and it is shown that the strongly $i$-bicritical graphs with independent domination number $i(G) \geq 5$ may be hard to characterize. It is shown …

The Distinguishing Number Of Some Special Kind Of Graphs, Arti Salat, Amit Sharma

Applications and Applied Mathematics: An International Journal (AAM)

In the present study, the distinguishing number of some different graphs is examined where different graphs like the coconut tree graph, firecracker graph, jellyfish graph, triangular book graph, and banana tree graph have been taken into account. The major goal of the proposed study is to understand the distinguishing number of different graphs for better insights. It is evident from the results that the distinguishing numbers and automorphism groups of the above-mentioned graphs have been carried out successfully.

Some Generalizations Of Corona Product Of Two Graphs, Aparajita Borah, Gajendra Pratap Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we are seeking to conceptualize the notion of corona product of two graphs to contrive some special types of graphs. That is, here our attempt is to regenerate a familiar graph as a product graph. We are considering seven familiar graphs here to reconstruct them with the help of corona product of two graphs. Such types of families of the graphs and operations can be used to study biological pathways as well as to find the optimal order and size for the special types of graphs.

Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

Rose-Hulman Undergraduate Mathematics Journal

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett

Journal of Humanistic Mathematics

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.

Recent Studies On The Super Edge-Magic Deficiency Of Graphs, Rikio Ichishima, Susana C. Lopez, Francesc Muntaner, Yukio Takahashi

Theory & Applications of Graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left(G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right) $. Also, $G$ is called super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $ \mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such …

A Survey Of Maximal K-Degenerate Graphs And K-Trees, Allan Bickle

Theory & Applications of Graphs

This article surveys results on maximal $k$-degenerate graphs, $k$-trees,

and related classes including simple $k$-trees, $k$-paths, maximal

outerplanar graphs, and Apollonian networks. These graphs are important

in many problems in graph theory and computer science. Types of results

surveyed include structural characterizations, enumeration, degree

sets and sequences, chromatic polynomials, algorithms, and related

extremal problems.

On The Singular Pebbling Number Of A Graph, Harmony R. Morris

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal …

The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, Toby Anderson

HMC Senior Theses

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on, Δ^(r,n), the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of Δ^(r,n). We compute its Euler characteristic and …

Solid Angle Measure Approximation Methods For Polyhedral Cones, Allison Fitisone

Theses and Dissertations--Mathematics

Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.

Ribando and Aomoto independently discovered a Taylor series formula for solid angle …