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| Presenter | Coauthors | Title & Abstract |
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Ethan Berkove Lafayette College |
Mike Brilleslyper United States Air Force Academy |
Subgraphs of Coprime Graphs on Sets of Consecutive Integers Let $\mathcal{A}_n^k$ denote the set of consecutive integers |
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Alvaro Carbonero University of Nevada Las Vegas |
Some properties of graphs with either convergent or divergent sequences of $P_n$-line graphs Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are adjacent in $G$ and lie in a common copy of $H$. For each nonnegative integer $k$, let $HL^{k+1}(G) = HL(HL^k(G))$ and $HL^0(G) = G$. We say that the sequence $\{ HL^k(G) \}$ converges if there exists a positive integer $N$ such that $k\geq N$ implies $HL^k(G) \cong HL^N(G)$. Finally, for $n \geq 3$ define $\Lambda_n$ as the set of all graphs $G$ whose sequence $\{HL^k(G) \}$ converges when $H\cong P_n$. The sets $\Lambda_3, \Lambda_4$ and $\Lambda_5$ have been characterized. To progress towards the characterization of $\Lambda_n$ in general, this presentation defines and studies the following property: a graph $G$ is minimally $n$-convergent if $G\in \Lambda_n$ but no proper subgraph of $G$ is in $\Lambda_n$. In addition, we present a way to study divergence, prove conditions that imply divergence, and use these conditions to develop some of the properties of minimally $n$-convergent graphs. |
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Braxton Carrigan Southern Connecticut State University |
John Asplund Metron, Inc. G. Green Southern Connecticut State University |
Skolem Number of Grid type graphs A Skolem sequence can be thought of as a labelled path where two vertices with the same positive integer label are that distance apart. This concept has naturally been generalized to graph labelling. We introduce a specific version called a proper Skolem labelling of a graph. This brings rise to the question; "what is the smallest set of consecutive positive integers we can use to proper Skolem label a graph?" This is known as the Skolem number of the graph. We will give the Skolem number of four natural vertex induced subgraphs of the triangular lattice graph. |
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Lisa Cenek Amherst College Brittany Gelb Muhlenberg College |
Interactive Theorem Proving with Lean The Lean proof assistant is a tool for verifying and generating proofs using a computer. In the past few years, mathematicians have started using Lean to formalize important results: Gusakov, Mehta, and Miller formalized the proof of Hall’s Marriage Theorem, and Kontorovich and Gomes formalized the statement of the Riemann Hypothesis. In this talk, we will introduce and demonstrate interactive theorem proving. We will share our motivation for learning about proof assistants and our experience getting started with Lean. We will also suggest potential topics in number theory and combinatorics that might be suitable for undergraduate research. |
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Ji Young Choi Shippensburg University of Pennsylvania |
Recurrence relations for the $b$-nomial numbers We consider the $(n,k)$-th binomial coefficient as the number of nonnegative integers, each of whose base-$2$ representation has $n$ digits and digit sum $k$. By using a different base $b$ and calculating the digit sum of a nonnegative integer or a multiple of $b-1$, we define four different types of generalizations of binomial coefficients. This talk will present the recurrence relations of each type. |
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Suchakree Chueluecha Lehigh University |
Tolson Bell Georgia Institute of Technology Lutz Warnke Georgia Institute of Technology |
The Sunflower Problem The goal of the sunflower problem is to find the smallest $r = r(p, k)$ such that every family of at least $r^k$ $k$-element sets must contain a sunflower with $p$ petals. Major breakthroughs by Alweiss-Lovett-Wu-Zhang and others show that $r = O(p \log(pk))$ suffices. In this talk, we present our improvement to $r = O(p \log(k)).$ |
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Peter Conley Lehigh University |
Acyclic Sum-list-coloring of Cylinders This talk will serve as an introduction to a variation of list coloring on a graph $G$ such that the sizes of the lists assigned to each vertex can be different. Furthermore, rather than giving a proper coloring, in which color classes induce independent subgraphs, we will discuss coloring $G$ such that the color classes induce acyclic subgraphs of $G$, as is done in finding vertex arboricity. The goal, then, is to minimize the sum of all list sizes such that $G$ is colorable regardless of the particular lists assigned. We will look at how achieving this goal is related to decycling sets, or feedback vertex sets, which leave acyclic graphs when removed. Finally, we will look specifically at achieving this goal for specific cartesian products of a cycle and a path. |
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Jacob DeCapua Ball State University |
Group-Based Sudoku Pair Latin Squares A Latin square of order $n$ is an $n \times n$ array coupled with $n$ symbols in which each row and column sees no repeated symbol. This property, often referred to as "Latin" has seen many generalizations. One such generalization is the Sudoku pair Latin square. And since groups have Latin structures appearing naturally as Cayley tables, focus is drawn to which groups can form these structures. Today a method to construct Sudoku pair Latin squares based on groups is presented and an understanding of where this research stands in the scope of Latin squares. |
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Josh Ducey James Madison University |
Abelian groups associated to strongly regular graphs Given any finite graph, there is an associated abelian group that arises from the graph’s Laplacian matrix. This "sandpile" group contains combinatorial information. For example, the order of the sandpile group of a connected graph is equal to the number of spanning trees of the graph. There has been much work in computing the sandpile groups of families of graphs. The class of strongly regular graphs has received recent attention. In this talk we present some new results on how the parameters of a strongly regular graph must constrain the structure of its sandpile group and other related groups. We discuss how these results apply to the existence question of strongly regular graphs, and we give some open research problems. |
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Ronald Gould Emory University |
Extensions Under Edge Density Conditions In this talk I will present several new results that extend well-known Theorems based on edge density conditions. These conditions include minimum degree, degree sums, and edge count. The types of results presented extend classic cycle results and one is a new and somewhat surprising subgraph result. |
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Nicholas Mayers Milwaukee School of Engineering |
Vincent Coll Lehigh University Nicholas Russoniello Lehigh University |
The breadth of nilpotent Lie poset algebras The breadth of a Lie algebra $L$ is defined to be the maximum of the dimensions of the images of $ad_x=[x,-]:L\to L$ for all $x\in L$. Here, we investigate the breadth of a class of Lie algebras defined by posets and called "nilpotent Lie poset algebras." In particular, for certain special families of posets, we find closed-form formulas for the breadth of the corresponding Lie algebras. |
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Michael Mossinghoff Center for Communications Research |
Shehzad Hathi UNSW Canberra Timothy Trudgian UNSW Canberra |
Wolstenholme and Vandiver primes In 1862 the Reverend J. Wolstenholme established a number of congruences modulo powers of primes for certain expressions involving binomial coefficients, harmonic numbers, and related quantities. A prime for which one of these congruences becomes a "super-congruence" (holding modulo a higher power of the prime) is known as a Wolstenholme prime. While it is conjectured that infinitely many such primes exist, very few are known. Wolstenholme primes can also be defined by using Bernoulli numbers, which arise widely in number theory, for example in the study of regular primes. A Vandiver prime can be defined in an analogous way, but using Euler numbers in place of Bernoulli numbers; these primes arose in the study of Fermat's Last Theorem. Here again it is conjectured that infinitely many exist, but only a few are known. We describe some new and extensive searches for Wolstenholme and Vandiver primes. To power this, we develop a number of new congruences for Bernoulli numbers and for Euler numbers that are favorable for computation, and we implement some highly parallel searches using GPUs. |
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Deniz Ozbay Lafayette College |
Coprime Labeling of Ladder Graphs A preprint by Ghorbani and Kamali provides an algorithm for the prime labeling of the ladder $P_n \times P_2$. In this talk, I will describe our work to generalize these results to a coprime labeling of $P_n \times P_2$, where the set of labels consists of the $2n$ positive integers starting at $k$. |
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Melea Roman Kansas State University |
Joshua Harrington Cedar Crest College Tony W. H. Wong Kutztown University |
$k$-fold Super Totient Numbers let $n$ be a positive integer and let $R(n)$ be the set of positive integers less than and relatively prime to $n$. If $R(n)$ can be partitioned into two subsets of equal sum then we say $n$ is super totient. This definition was introduced in 2017 by Mahmood and Ali and then completely classified in 2019 by Harrington and Wong. The generalization of this concept into $k$ subsets of equal sum we call $k$-fold super totient numbers. We give complete classifications for $3$-fold, $5$-fold and show that for any prime $k$, the classification of all $k$-fold super totient numbers can be found computationally. |
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Linds Wise Appalachian State University |
Cameron Byer Eastern Mennonite University Tyler Dvorachek University of Wisconsin-Green Bay Emily Eckard Mount St. Mary's University Joshua Harrington Cedar Crest College Tony W. H. Wong Kutztown University |
On the Properties of Fibotomic Polynomials, I Fibonacci polynomials are defined recursively in the following manner: $U_0(x) = 0$ and $U_1(x) = 1$, and for all $n \geq 2$, $U_n(x) = U_{n-2}(x) + xU_{n-1}(x)$. In this talk, we consider the irreducible factors of Fibonacci polynomials, which are called the Fibotomic polynomials. The Fibotomic polynomials are known to share similar root structures with cyclotomic polynomials, which makes them an especially interesting class of polynomials to study. We prove several analogous properties for the Fibotomic polynomials that are well-known for the cyclotomic polynomials, including the factorization of Fibotomic polynomials modulo a prime. |
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Wing Hong Tony Wong Kutztown University of Pennsylvania |
Cameron Byer Eastern Mennonite University Tyler Dvorachek University of Wisconsin-Green Bay Emily Eckard Mount St. Mary's University Joshua Harrington Cedar Crest College Lindsey Wise Appalachian State University |
On the properties of Fibotomic polynomials, II Fibotomic polynomials are defined as in the talk "On the properties of Fibotomic polynomials, I." In this talk, we are going to present the discriminant and resultant of Fibotomic polynomials. Despite the result is closely related to number theory, techniques involved rely heavily on trigonometric identities and combinatorics. |