Wang Tilings In Arbitrary Dimensions, 2024 Oregon State University

#### 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, 2024 University of Victoria

#### 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 …

Some Generalizations Of Corona Product Of Two Graphs, 2024 National Institute of Technology, Sikkim, India

#### 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.

The Distinguishing Number Of Some Special Kind Of Graphs, 2024 Shri P.N. Pandya Arts, M.P. Pandya Science and Smt. D.P. Pandya Commerce College

#### 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.

Optimizing Buying Strategies In Dominion, 2024 Georgia Southern University

#### 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, 2024 Sheffield Hallam University

#### 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, 2024 Kokushikan University

#### 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, 2024 Purdue University

#### 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, 2024 University of Waterloo

#### 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, 2024 Georgia Southern University

#### 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.

Reducing Food Scarcity: The Benefits Of Urban Farming, 2023 Brigham Young University

#### Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

*Journal of Nonprofit Innovation*

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, 2023 Seikei University

#### Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, Kengo Enami, Yumiko Ohno

*Theory and Applications of Graphs*

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.

For two triangulations $G$ and $G'$ on a surface, $\psi_3(G)$ may not be equal to $\psi_3(G')$ even if $G$ …

The Ricci Curvature On Simplicial Complexes, 2023 Shimane University

#### The Ricci Curvature On Simplicial Complexes, Taiki Yamada

*Theory and Applications of Graphs*

We define the Ricci curvature on simplicial complexes modifying the definition of the Ricci curvature on graphs, and prove upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial complexes by the Ricci curvature.

Toughness Of Recursively Partitionable Graphs, 2023 University of Vermont

#### Toughness Of Recursively Partitionable Graphs, Calum Buchanan, Brandon Du Preez, K. E. Perry, Puck Rombach

*Theory and Applications of Graphs*

A simple graph *G = (V,E)* on *n* vertices is said to be recursively partitionable (RP) if *G ≃ K1*, or if *G* is connected and satisfies the following recursive property: for every integer partition *a _{1}, a_{2}, . . . , a_{k}* of

*n*, there is a partition

*{A*of

_{1},A_{2}, . . . ,A_{k}}*V*such that each

*|A*, and each induced subgraph

_{i}| = a_{i}*G[A*is RP

_{i}]*(1 ≤ i ≤ k)*. We show that if

*S*is a …

Wiener Index In Graphs Given Girth, Minimum, And Maximum Degrees, 2023 Alabama State University, USA

#### Wiener Index In Graphs Given Girth, Minimum, And Maximum Degrees, Fadekemi J. Osaye, Liliek Susilowati, Alex S. Alochukwu, Cadavious Jones

*Theory and Applications of Graphs*

Let $G$ be a connected graph of order $n$. The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ by Kouider and Winkler \cite{Kouider} was improved significantly by Alochukwu and Dankelmann \cite{Alex} for graphs containing a vertex of large degree $\Delta$ to $W(G) \leq {n-\Delta+\delta \choose 2} \big( \frac{n+2\Delta}{\delta+1}+4 \big)$. In this paper, we give upper bounds on the Wiener index of $G$ in terms of order …

On The Hardness Of The Balanced Connected Subgraph Problem For Families Of Regular Graphs, 2023 Indian Institute of Technology Hyderabad

#### On The Hardness Of The Balanced Connected Subgraph Problem For Families Of Regular Graphs, Harsharaj Pathak

*Theory and Applications of Graphs*

The Balanced Connected Subgraph problem (BCS) was introduced by Bhore et al. In the BCS problem we are given a vertex-colored graph G = (V, E) where each vertex is colored “red” or “blue”. The goal is to find a maximum cardinality induced connected subgraph H of G such that H contains an equal number of red and blue vertices. This problem is known to be NP-hard for general graphs as well as many special classes of graphs. In this work we explore the time complexity of the BCS problem in case of regular graphs. We prove that the BCS …

The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, 2023 William & Mary

#### The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

*Cybersecurity Undergraduate Research Showcase*

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

A Bridge Between Graph Neural Networks And Transformers: Positional Encodings As Node Embeddings, 2023 East Tennessee State University

#### A Bridge Between Graph Neural Networks And Transformers: Positional Encodings As Node Embeddings, Bright Kwaku Manu

*Electronic Theses and Dissertations*

Graph Neural Networks and Transformers are very powerful frameworks for learning machine learning tasks. While they were evolved separately in diverse fields, current research has revealed some similarities and links between them. This work focuses on bridging the gap between GNNs and Transformers by offering a uniform framework that highlights their similarities and distinctions. We perform positional encodings and identify key properties that make the positional encodings node embeddings. We found that the properties of expressiveness, efficiency and interpretability were achieved in the process. We saw that it is possible to use positional encodings as node embeddings, which can be …

Zero-Knowledge Reductions And Confidential Arithmetic, 2023 Clemson University

#### Zero-Knowledge Reductions And Confidential Arithmetic, Marvin Jones

*All Dissertations*

The changes in computing paradigms to shift computations to third parties have resulted in the necessity of these computations to be provable. Zero-knowledge arguments are probabilistic arguments that are used to to verify computations without secret data being leaked to the verifying party.

In this dissertation, we study zero-knowledge arguments with specific focus on reductions. Our main contributions are:

- Provide a thorough survey in a variety of zero-knowledge techniques and protocols.
- Prove various results of reductions that can be used to study interactive protocols in terms of subroutines. Additionally, we identify an issue in the analogous definition of zero-knowledge for …

Generalized Vulnerability Measures Of Graphs, 2023 Clemson University

#### Generalized Vulnerability Measures Of Graphs, Julia Vanlandingham

*All Theses*

Several measures of vulnerability of a graph look at how easy it is to disrupt the network by removing/disabling vertices. As graph-theoretical parameters, they treat all vertices alike: each vertex is equally important. For example, the integrity parameter considers the number of vertices removed and the maximum number of vertices in a component that remains. We consider the generalization of these measures of vulnerability to weighted vertices in order to better model real-world applications. In particular, we investigate bounds on the weighted versions of connectivity and integrity, when polynomial algorithms for computation exist, and other characteristics of the generalized measures.