Physical Sciences and Mathematics Commons^™

Graph And Group Theoretic Properties Of The Soma Cube And Somap, Kyle Asbury, Ben Glancy

Mathematical Sciences Technical Reports (MSTR)

The SOMA Cube is a puzzle toy in which seven irregularly shaped blocks must be fit together to build a cube. There are 240 distinct solutions to the SOMA Cube. One rainy afternoon, Conway and Guy created a graph of all the solutions by manually building each solution. They called their graph the SOMAP. We studied how the geometric structure of the SOMA Cube pieces informs the graph theoretic properties of the SOMAP, such as subgraphs that can or cannot appear and vertex centrality. We have also used permutation group theory to decipher notation used by Knuth in previous work …

The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve

Electronic Thesis and Dissertation Repository

Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.

In this thesis, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us …

Cohen-Macaulay Type Of Open Neighborhood Ideals Of Unmixed Trees, Jounglag Lim

All Theses

Given a tree T and a field k, we define the open neighborhood ideal N(T) of T in k[V] to be the ideal generated by the open neighborhoods of all vertices in the graph. If T is unmixed with respect to the total domination problem, then it is known that N(T) is Cohen-Macaulay. Our goal is to compute the (Cohen-Macaulay) type of k[V]/N(T) using graph theoretical properties of T. We achieve this by using homological algebra and properties of monomial ideals. Along the way, we also provide a different characterization of unmixed trees and a generalization of the total dominating …

Modeling Virus Diffusion On Social Media Networks With The Smirq Model, Justin Browning, Arnav Mazumder, Gowri Nanda

Rose-Hulman Undergraduate Mathematics Journal

As social networking services become more complex and widespread, users become increasingly susceptible to becoming infected with malware and risk their data being compromised. In the United States, it costs the government billions of dollars annually to handle malware attacks. Additionally, computer viruses can be spread through schools, businesses, and individuals’ personal devices and accounts. Malware affecting larger groups of people causes problems with privacy, personal files, and financial security. Thus, we developed the probabilistic SMIRQ (pSMIRQ) model that shows how a virus spreads through a generated network as a way to track and prevent future viruses. Our model is …

Big Two And N-Card Poker Probabilities, Brian Wu, Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of n≥5. In particular, we derive closed expressions for the probabilities of flush, straight and full house and show that the probability of a flush is less than a straight …

Penney’S Game For Permutations, Yixin Lin

Dartmouth College Ph.D Dissertations

We explore the permutation analog of Penney's game for coin flips. Two players, in order, each choose a permutation of length $k\ge3$. Then a sequence of independent random values from a continuous distribution is generated until the relative order of the last $k$ numbers matches one of the chosen permutations, declaring the player who selected that permutation as the winner.

We calculate the winning probabilities for all pairs of permutations of length $3$ and some pairs of length $4$, demonstrating the non-transitive property of this game, consistent with the original word version. Alternatively, we provide formulas for computing the winning …

On Pattern Avoidance And Dynamical Algebraic Combinatorics, Benjamin Adenbaum

Dartmouth College Ph.D Dissertations

Over the past decade since the term `dynamical algebraic combinatorics' was coined there has been a tremendous amount of activity in the field. Adding to that growing body of work this thesis hopes to be a step towards a broader study of pattern avoidance within dynamical algebraic combinatorics and helps initiate that by considering an action of rowmotion on 321-avoiding permutations. Additionally within we show the first known instance of piecewise-linear rowmotion periodicity for an infinite family of posets that does not follow from a more general birational result. Finally we show that the code of permutation restricted to permutations …

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

Student Research Submissions

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

An Alternate Proof For The Top-Heavy Conjecture On Partition Lattices Using Shellability, Brian Macdonald, Josh Hallam

Honors Thesis

A partially ordered set, or poset, is governed by an ordering that may or may not relate
any pair of objects in the set. Both the bonds of a graph and the partitions of a set are
partially ordered, and their poset structure can be depicted visually in a Hasse diagram. The
partitions of {1, 2, ..., n} form a particularly important poset known as the partition lattice
Πn. It is isomorphic to the bond lattice of the complete graph Kn, making it a special case
of the family of bond lattices.
Dowling and Wilson’s 1975 Top-Heavy Conjecture states that …

Approval Gap Of Weighted K-Majority Tournaments, Jeremy Coste, Breeann Flesch, Joshua D. Laison, Erin Mcnicholas, Dane Miyata

Theory and Applications of Graphs

A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in $T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$ weighted by the number of voters preferring $u$.

We define the …

The Forget Time For Random Walks On Trees Of A Fixed Diameter, Lola R. Vescovo

Mathematics, Statistics, and Computer Science Honors Projects

A mixing measure is the expected length of a random walk on a graph given a set of starting and stopping conditions. We study a mixing measure called the forget time. Given a graph G, the pessimal access time for a target distribution is the expected length of an optimal stopping rule to that target distribution, starting from the worst initial vertex. The forget time of G is the smallest pessimal access time among all possible target distributions. We prove that the balanced double broom maximizes the forget time on the set of trees on n vertices with diameter …

Asteroidal Sets And Dominating Targets In Graphs, Oleksiy Al-Saadi

Department of Computer Science and Engineering: Dissertations, Theses, and Student Research

The focus of this PhD thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the …

Domination In Graphs And The Removal Of A Matching, Geoffrey Boyer

All Theses

We consider how the domination number of an undirected graph changes on the removal of a maximal matching. It is straightforward that there are graphs where no matching removal increases the domination number, and where some matching removal doubles the domination number. We show that in a nontrivial tree there is always a matching removal that increases the domination number; and if a graph has domination number at least $2$ there is always a maximal matching removal that does not double the domination number. We show that these results are sharp and discuss related questions.

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 …

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 and 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 and 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 and 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 …

Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron

Honors College Theses

The properties of n-color compositions have been studied parallel to those of regular compositions. The conjugate of a composition as defined by MacMahon, however, does not translate well to n-color compositions, and there is currently no established analogous concept. We propose a conjugation rule for cyclic n-color compositions. We also count the number of self-conjugates under these rules and establish a couple of connections between these and regular compositions.

An Approach To Multidimensional Discrete Generating Series, Svetlana S. Akhtamova, Tom Cuchta, Alexander P. Lyapin

Mathematics Faculty Research

We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case.

Symmetry And Structures Of Apn Functions And Sidon Sets, Darrion Thornburgh

Senior Projects Spring 2024

Let $\mathbb{F}_p^n$ be the $n$-dimensional vector space over $\mathbb{F}_p$. The graph $\mathcal{G}_F = \{ (x, F(x)) : x \in \mathbb{F}_p^n \}$ of a vectorial function $F \colon \mathbb{F}_p^n \to \mathbb{F}_p^m$ can have interesting combinatorial properties depending on varying cryptographic conditions on $F$. A vectorial Boolean function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ is almost perfect nonlinear (APN) if there are at most $2$ solutions to the equation $F(x+a) + F(x) = b$ for all $a,b \in \mathbb{F}_2^n$ where $a \neq 0$. Equivalently, $F$ is APN if and only if $\mathcal{G}_F$ is a Sidon set, that is, a set in $\mathbb{F}_2^n$ where …

Folding And Embedding Cubical Complexes, Skye Rothstein

Senior Projects Spring 2024

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

In this project we study the folding properties of several special classes of cubical complexes. First, we look at polyominoids, which are arrangements of congruent squares in 3-space, glued edge-to-edge at 90° and 180° angles. We construct and analyze the mechanical configuration space for n-cell polyominoids, which is a graph with vertex set given by all n-cell polyominoids, where two vertices are connected by an edge if you can transform one into the other by one hinge movement. For n = 4, we provide a complete …

Problems In Chemical Graph Theory Related To The Merrifield-Simmons And Hosoya Topological Indices, William B. O'Reilly

Electronic Theses and Dissertations

In some sense, chemical graph theory applies graph theory to various physical sciences. This interdisciplinary field has significant applications to structure property relationships, as well as mathematical modeling. In particular, we focus on two important indices widely used in chemical graph theory, the Merrifield-Simmons index and Hosoya index. The Merrifield-Simmons index and the Hosoya index are two well-known topological indices used in mathematical chemistry for characterizing specific properties of chemical compounds. Substantial research has been done on the two indices in terms of enumerative problems and extremal questions. In this thesis, we survey known extremal results and consider the generalized …

On Graph Decompositions And Designs: Exploring The Hamilton-Waterloo Problem With A Factor Of 6-Cycles And Projective Planes Of Order 16, Zazil Santizo Huerta

Dissertations, Master's Theses and Master's Reports

This dissertation tackles the challenging graph decomposition problem of finding solutions to the uniform case of the Hamilton-Waterloo Problem (HWP). The HWP seeks decompositions of complete graphs into cycles of specific lengths. Here, we focus on cases with a single factor of 6-cycles. The dissertation then delves into the construction of 1-rotational designs, a concept from finite geometry. It explores the connection between these designs and finite projective planes, which are specific geometric structures. Finally, the dissertation proposes a potential link between these seemingly separate areas. It suggests investigating whether 1-rotational designs might hold the key to solving unsolved instances …