Mathematics Commons^™

The Logic Of Leaves: Mathematical Poems And Graphs, Ksawery Tomczak

Journal of Humanistic Mathematics

This collection explores the intersection of mathematics and poetry through six original works that blend formal rigor with lyrical expression. Each poem draws from a distinct mathematical concept—proof, set theory, combinatorics, graph theory, and cardinality—reframing it through metaphoric and aesthetic lenses. From the lyrical contemplation of infinite sets to the emotive longing of a graph in love, the verses reveal the emotional resonance and philosophical depth embedded in mathematical thought. The collection culminates in a visual piece rendered in \LaTeX and TikZ, illustrating all integer partitions of the number five as a branching tree, accompanied by a poem that sings …

Hyper Face-Magic Graphs, Ross Belgram, Donald Mcginn

Theory & Applications of Graphs

For a planar graph G of order n, let F(G) be the set of all faces of G embedded into R 2 , including the exterior face. A bijective vertex labeling f : V (G) → {1, 2, ..., n} induces a face labeling f ∗ : F(G) → N defined by setting f ∗ (F) equal to the sum of all labels of the boundary vertices of F. The graph G is said to be hyper face-magic if there exists a vertex labeling whose induced face labeling is constant. In this paper, we state properties of hyper face-magic graphs …

From Total Domination To Graph Coloring, Sawyer Isaac Osborn

Dissertations

A question involving a chess piece called a prince on the 8×8 chessboard leads to a concept in graph theory involving total domination. We say a vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A subset S of the vertex set of a graph G is a total dominating set for G if every vertex in G is totally dominated by at least one vertex of S. If S is a total dominating set of G, then σ_S(v) denotes the number of …

Rainbow Dominating Sets Of Graphs, Samuel L. Powell

All Graduate Theses and Dissertations, Fall 2023 to Present

The content of this thesis may be compared to a stack of plates with a common design that are broken, one at a time. If the plates are broken into large enough pieces you would be able to choose one fragment from each broken plate to discover the entire design that the plates share. For example, if you were still missing the center of the design after taking a piece from some plates, you could look for the piece of the next broken plate with the region in question.

We study the question of how many broken plates might guarantee …

Graph Theoretical Modeling Of Self-Assembling Dna Of The Double Cone Graph, Philiffe Tebalan, Evan Burns

Rose-Hulman Undergraduate Mathematics Journal

The unique properties of double-stranded DNA molecules make DNA a valuable structural material with which to form nanostructures, and the field of DNA nanotechnology is largely based on this premise. By modeling nanostructures with discrete graphs, efficient DNA self-assembly becomes a mathematical puzzle. These nanostructures have wide-ranging applications, such as containers for the transport and release of nano-cargos, templates for the controlled growth of nano-objects, and in drug-delivery methods. This research centers around exploring graph theoretical and combinatorial properties of DNA self-assembly to optimize the nanostructure construction for the Double Cone Graph.

Spectral Decimation On The Schreier Graphs Of The Basilica Group: A Thesis In Fractal Analysis, Anne D. Bannon

Scripps Senior Theses

This thesis is intended to provide a comprehensive overview of the literature required to fully understand research conducted during the University of Connecticut's Fractals & Stochastics REU in the summer of 2025. The literature review includes a description of Robert Strichartz's seminal work pertaining to the Laplacian spectrum of the Sierpiński Gasket, which provides a framework for how we approach studying the spectrum of the basilica Julia set. Defining the basilica Julia set and the closely-related Basilica group involves graph theory, automata theory, iterated monodromy group theory, and amenable group theory. Further time is dedicated to defining the graph Laplacian …

Likelihood-Based Inference For Random Networks With Changepoints, Daniel Cirkovic, Tiandong Wang, Xianyang Zhang

Mathematical and Statistical Science Faculty Research and Publications

Generative, temporal network models play an important role in analyzing the dependence structure and evolution patterns of complex networks. Due to the complicated nature of real network data, it is often naive to assume that the underlying data-generative mechanism itself is invariant with time. Such observation leads to the study of changepoints or sudden shifts in the distributional structure of the evolving network. In this paper, we propose a likelihood-based methodology to detect changepoints in undirected, affine preferential attachment networks where, upon introduction, a new node selects one old to attach to with probability proportional to its degree. In particular, …

Dominating Hadwiger's Conjecture For 2k2-Free Graphs, Thomas Tibbetts

Honors Undergraduate Theses

A dominating K_t minor in a graph �� is a sequence (��₁,…,��_t) of pairwise disjoint non-empty connected subgraphs of ��, such that for 1≤��< ��≤��, every vertex in ��_j has a neighbor in ��_i. Replacing “every vertex in ��_j” by “some vertex in ��_j” retrieves the standard definition of a ��_t minor. The strengthened notion was introduced by Illingworth and Wood in 2024, who asked whether every graph with chromatic number �� contains a dominating ��_t minor. This is a substantial strengthening of the celebrated Hadwiger’s Conjecture, which asserts that every …

A Comprehensive Study Of Clique Graphs And Clique Regular Graphs, Connor Phillips

Senior Honors Projects, 2020-current

If Γ is a graph for which every edge is in exactly one clique of order ω, then one can form a new graph with vertex set equal to these cliques.  This is a generalization of the line graph of Γ.  We discover many general results and classifications related to these clique graphs that will be useful to researchers studying graphs with this property. In particular, we find bounds on the spectrum of Γ (with exact results when Γ is k-regular) and some complete classifications when Γ is strongly regular. We apply our results to derive novel information about the …

Robustness Of Network Inference Algorithms Under Network Class Misspecification, Roberto Ceja

Electronic Theses, Projects, and Dissertations

Inferring phylogenies in the presence of hybridization remains a difficult problem. As a result, many current methods for reconstructing phylogenetic networks are restricted to a simple class of networks known as level-1. This restriction arises from theoretical considerations rather than empirical evidence, with real data possibly arising from complex networks. In this work, we evaluate the robustness of two level-1 network inference methods, SNaQ and NANUQ+, through a simulation study, when the input data originates from a more complex network. Specifically, we investigate whether these methods can accurately recover important features of the true species network, such as the circular …

On Variations Of Isolation In Graphs, Geoffrey Boyer

All Dissertations

In 2015, Caro and Hansberg introduced a wonderful and natural generalization of the well-studied parameter of domination in graphs. For a graph $G$ and a family of graphs $\FF$, they define $S$ to be an $\FF$-isolating set if $G-N[S]$ contains no member of $\FF$ as a subgraph. The case where $\FF=\{K_1\}$ coincides with domination. The case where $\FF=\{K_2\}$ is now simply referred to as an isolating set. We strengthen known results about the isolation number of a graph, and explore variations of the parameter including the independent and total versions.

In particular for connected graphs of order $n$, a bound …

Structural And Computational Results On Equitably Dissectible Graphs, Ann Wells Clifton

Doctoral Dissertations

This dissertation introduces the class of equitably dissectible graphs — graphs whose vertex set can be recursively partitioned into two parts whose sizes differ by at most one where the two induced subgraphs are connected. In Chapter 2, we show that a graph is equitably dissectible if and only if it has a spanning tree that is equitably dissectible, and we use this result to establish connections between the class of equitably dissectible graphs and Hamiltonian graphs. We also fully characterize the equitably dissectible spiders of the form S3(a) and S3(a, b), identifying a family of unruly spiders that are …

Unavoidable Immersions And Topological Minors Of K-Edge-Connected Graphs, Brittian Qualls

LSU Doctoral Dissertations

In this work, we present results concerning the unavoidable structures in large and infinite k-edge-connected graphs. These results are inspired by the classical result of Ramsey, who proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either K_r or $\overline{K_r}$. We consider different graph containment relations, focusing primarily on the immersion relation. In the case of finite graphs, we provide the unavoidable immersions of 4-edge-connected graphs and prove that linear edge-connectivity suffices to immerse the graph C_t,r.

This dissertation also considers infinite graphs. We present the …

Construction Techniques For Linear Realizations Of Multisets With Small Support, M. A. Ollis

Emerson Authors, Researchers, & Creators

The Buratti-Horak Rosa Conjecture is a problem in graph theory asking when we can find a Hamiltonian path in the complete graph that has various properties to do with edge lengths. In this paper, we use constructive methods to show that the conjecture is true for various parameter sets where it was previously unknown.

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 …

The B-Chromatic Number Of Super Cycles, Blair E. Johnson

DePaul Discoveries

The b-chromatic number, a characteristic studied in graph theory, was introduced less than 25 years ago, and while some statistics have been gathered of its properties, the b-chromatic number has yet to be explored in many graphs and circumstances. Before the Undergraduate Summer Research Program, I had no exposure to graph theory, let alone b-colorings. Throughout the program, I developed a fundamental understanding of the subject and began to pursue my own research inquiries. While this objective was difficult to attain at the beginning, I learned strategies and methods to hone my ability to delve into different graphs, and soon …

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

School of Computing: 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 …

Optimal Network Analysis Through Vertex Order Coloring Of Intuitionistic Fuzzy Graph Operations, A. Meenakshi, S. Dhanushiya, Hong Qin, Maniyandy Elangovan

Data Science Faculty Publications

Intuitionistic fuzzy graphs IFGs are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real-world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring IFVOC and analyzed the alpha-strong (alpha str), beta-strong (beta str), and gamma-strong (gamma str) vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of alpha str vertices, the chromatic number, and the weight …

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.

Combinatorially Orthogonal Paths, Sean A. Bailey, David E. Brown, Leroy Beaseley

Communications on Number Theory and Combinatorial Theory

Vectors x=(x₁,x₂,...,x_n)^T and y=(y₁,y₂,...,y_n)^T are combinatorially orthogonal if |{i:x_iy_i≠0}|≠1. An undirected graph G=(V,E) is a combinatorially orthogonal graph if there exists f:V→ℝⁿ such that for any u,v∈V, uv∉E iff f(u) and f(v) are combinatorially orthogonal. We will show that every graph has a combinatorially orthogonal representation. We will show …

The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales

Rose-Hulman Undergraduate Mathematics Journal

DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas …

A Survey On Online Matching And Ad Allocation, Ryan Lee

Theses

One of the classical problems in graph theory is matching. Given an undirected graph, find a matching which is a set of edges without common vertices. In 1990s, Richard Karp, Umesh Vazirani, and Vijay Vazirani would be the first computer scientists to use matchings for online algorithms [8]. In our domain, an online algorithm operates in the online setting where a bipartite graph is given. On one side of the graph there is a set of advertisers and on the other side we have a set of impressions. During the online phase, multiple impressions will arrive and the objective of …

Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance

Mathematics, Statistics, and Computer Science Honors Projects

A mixing measure is the expected length of a random walk in a graph given a set of starting and stopping conditions. We determine the tree structures of order n with diameter d that minimize and maximize for a few mixing measures. We show that the maximizing tree is usually a broom graph or a double broom graph and that the minimizing tree is usually a seesaw graph or a double seesaw graph.

Exploring The Structure Of Partial Difference Sets With Denniston Parameters, Nicolas Ferree

Honors Theses

In this work, we investigate the structure of particular partial difference sets (PDS) of size 70 with Denniston parameters in an elementary abelian group and in a nonelementary abelian group. We will make extensive use of character theory in our investigation and ultimately seek to understand the nature of difference sets with these parameters. To begin, we will cover some basic definitions and examples of difference sets and partial difference sets. We will then move on to some basic theorems about partial difference sets before introducing a group ring formalism and using it to explore several important constructions of partial …

Zonality In Graphs, Andrew Bowling

Dissertations

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …

Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon

Senior Theses

This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

Iterated Jump Graphs, Fran Herr, Legrand Jones Ii

Rose-Hulman Undergraduate Mathematics Journal

The jump graph J(G) of a simple graph G has vertices which represent edges in G where two vertices in J(G) are adjacent if and only if the corresponding edges in G do not share an endpoint. In this paper, we examine sequences of graphs generated by iterating the jump graph operation and characterize the behavior of this sequence for all initial graphs. We build on work by Chartrand et al. who showed that a handful of jump graph sequences terminate and two sequences converge. We extend these results by showing that there are no non-trivial repeating sequences of jump …

The Chromatic Index Of Ring Graphs, Lilian Shaffer

Rose-Hulman Undergraduate Mathematics Journal

The goal of graph edge coloring is to color a graph G with as few colors as possible such that each edge receives a color and that adjacent edges, that is, different edges incident to a common vertex, receive different colors. The chromatic index, denoted χ′(G), is the minimum number of colors required for such a coloring to be possible. There are two important lower bounds for χ′(G) on every graph: maximum degree, denoted ∆(G), and density, denoted ω(G). Combining these two lower bounds, we know that every graph’s chromatic index must be at least ∆(G) or …