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Harmonious Labelings Via Cosets And Subcosets, Jared L. Painter, Holleigh C. Landers, Walker M. Mattox
Harmonious Labelings Via Cosets And Subcosets, Jared L. Painter, Holleigh C. Landers, Walker M. Mattox
Theory and Applications of Graphs
In [Abueida, A. and Roblee, K., More harmonious labelings of families of disjoint unions of an odd cycle and certain trees, J. Combin. Math. Combin. Comput., 115 (2020), 61-68] it is shown that the disjoint union of an odd cycle and certain paths is harmonious, and that certain starlike trees are harmonious using properties of cosets for a particular subgroup of the integers modulo m, where m is the number of edges of the graph. We expand upon these results by first exploring the numerical properties when adding values from cosets and subcosets in the integers modulo m. …
Structure Of Number Theoretic Graphs, Lee Trent
Structure Of Number Theoretic Graphs, Lee Trent
Mathematical Sciences Technical Reports (MSTR)
The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …
Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis
Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis
Student Research
While doing research looking for ways to categorize causal networks generated by Sequential Substitution Systems, I created a new notation to compactly summarize concatenations of integers or strings of integers, including infinite sequences of these, in the same way that sums, products, and unions of sets can be summarized. Using my method, any sequence of integers or strings of integers with a closed-form iterative pattern can be compactly summarized in just one line of mathematical notation, including graphs generated by Sequential Substitution Systems, many Primitive Pythagorean Triplets, and various Lucas sequences including the Fibonacci sequence and the sequence of square …
3-Uniform 4-Path Decompositions Of Complete 3-Uniform Hypergraphs, Rachel Mccann
3-Uniform 4-Path Decompositions Of Complete 3-Uniform Hypergraphs, Rachel Mccann
Mathematical Sciences Undergraduate Honors Theses
The complete 3-uniform hypergraph of order v is denoted as Kv and consists of vertex set V with size v and edge set E, containing all 3-element subsets of V. We consider a 3-uniform hypergraph P7, a path with vertex set {v1, v2, v3, v4, v5, v6, v7} and edge set {{v1, v2, v3}, {v2, v3, v4}, {v4, v5, v6}, {v5, v6 …
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
The Review: A Journal of Undergraduate Student Research
The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …
Minimality Of Integer Bar Visibility Graphs, Emily Dehoff
Minimality Of Integer Bar Visibility Graphs, Emily Dehoff
University Honors Theses
A visibility representation is an association between the set of vertices in a graph and a set of objects in the plane such that two objects have an unobstructed, positive-width line of sight between them if and only if their two associated vertices are adjacent. In this paper, we focus on integer bar visibility graphs (IBVGs), which use horizontal line segments with integer endpoints to represent the vertices of a given graph. We present results on the exact widths of IBVGs of paths, cycles, and stars, and lower bounds on trees and general graphs. In our main results, we find …
Intersection Cographs And Aesthetics, Robert Haas
Intersection Cographs And Aesthetics, Robert Haas
Journal of Humanistic Mathematics
Cographs are complete graphs with colored lines (edges); in an intersection cograph, the points (vertices) and lines (edges) are labeled by sets, and the line between each pair of points is (or represents) their intersection. This article first presents the elementary theory of intersection cographs: 15 are possible on 4 points; constraints on the triangles and quadrilaterals; some forbidden configurations; and how, under suitable constraints, to generate the points from the lines alone. The mathematical theory is then applied to aesthetics, using set cographs to describe the experience of a person enjoying a picture (Mu Qi), poem (Dickinson), play (Shakespeare), …
Local-Global Results On Discrete Structures, Alexander Lewis Stevens
Local-Global Results On Discrete Structures, Alexander Lewis Stevens
Electronic Theses and Dissertations
Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …
Matrix Interpretations And Tools For Investigating Even Functionals, Benjamin Stringer
Matrix Interpretations And Tools For Investigating Even Functionals, Benjamin Stringer
Theses and Dissertations--Computer Science
Even functionals are a set of polynomials evaluated on the terms of hollow symmetric matrices. Their properties lend themselves to applications such as counting subgraph embeddings in generic (weighted or unweighted) host graphs and computing moments of binary quadratic forms, which occur in combinatorial optimization. This research focuses primarily on counting subgraph embeddings, which is traditionally accomplished with brute-force algorithms or algorithms curated for special types of graphs. Even functionals provide a method for counting subgraphs algebraically in time proportional to matrix multiplication and is not restricted to particular graph types. Counting subgraph embeddings can be accomplished by evaluating a …
Equitable Coloring Of Complete Tripartitle Graphs, Maxwell Vlam, Bailey Orehosky, Dominic Ditizio
Equitable Coloring Of Complete Tripartitle Graphs, Maxwell Vlam, Bailey Orehosky, Dominic Ditizio
Capstone Showcase
In this paper, we prove the Equitable Coloring Conjecture for variations of complete tripartite graphs with graphs K_n,n,n, K_n,n,2n, K_n,n,n+2, and K_n,n+2,n+4.