Discrete Mathematics and Combinatorics Commons^™

Pbw Bases And Marginally Large Tableaux In Type D, Ben Salisbury, Adam Schultze, Peter Tingley

Mathematics and Statistics: Faculty Publications and Other Works

We give an explicit description of the unique crystal isomorphism between two realizations of B(∞) in type D: that using marginally large tableaux and that using PBW monomials with respect to one particularly nice reduced expression of the longest word

Sparse Trees With A Given Degree Sequence, Ao Shen

College of Graduate Studies: Theses & Dissertations

In this thesis, we consider the properties of sparse trees and summarized a certain class of trees under some constraint (including with a given degree sequence, with given number of leaves, with given maximum degree, etc.) which have maximum Wiener index and the minimum number of subtrees at the same time. Wiener index is one of the most important topological indices in chemical graph theory. Steiner k�� Wiener index can be regarded as the generalization of Wiener index, when k = 2, Steiner Wiener index is the same as Wiener index. Steiner k�� Wiener index of a tree T is …

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

Murray State Theses and Dissertations

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as filling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is …

On The Domination Chain Of M By N Chess Graphs, Kathleen Johnson

Murray State Theses and Dissertations

A survey of the six domination chain parameters for both square and rectangular chess boards are discussed.

Self-Assembly Of Dna Graphs And Postman Tours, Katie Bakewell

UNF Graduate Theses and Dissertations

DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and …

On The Density Of The Odd Values Of The Partition Function, Samuel Judge

Dissertations, Master's Theses and Master's Reports

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable …

Notions Of Rough Neutrosophic Digraphs, Florentin Smarandache, Nabeela Ishfaq, Sidra Sayed, Muhammad Akram

Branch Mathematics and Statistics Faculty and Staff Publications

Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we apply the concept of rough NS theory to graphs and present a new kind of graph structure, rough neutrosophic digraphs. We present certain operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also present an application of a rough neutrosophic digraph in decision-making.

Spanning Tree Problem With Neutrosophic Edge Weights, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea, Arindam Dey, Le Hoang Son

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set and neutrosophic logic theory are renowned theories to deal with complex, not clearly explained and uncertain real life problems, in which classical fuzzy sets/models may fail to model properly. This paper introduces an algorithm for finding minimum spanning tree (MST) of an undirected neutrosophic weighted connected graph (abbr. UNWCG) where the arc/edge lengths are represented by a single valued neutrosophic numbers. To build the MST of UNWCG, a new algorithm based on matrix approach has been introduced. The proposed algorithm is compared to other existing methods and finally a numerical example is provided.

Subset Vertex Graphs For Social Networks, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. We have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and …

Mod Rectangular Natural Neutrosophic Numbers, Florentin Smarandache, K. Ilanthenral, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of MOD rectangular planes. The functions on them behave very differently when compared to MOD planes (square). These are different from the usual MOD planes. Algebraic structures on these MOD rectangular planes are defined and developed. However we have built only MOD interval natural neutrosophic products

Generalized Matching Preclusion In Bipartite Graphs, Zachary Wheeler, Eddie Cheng, Dana Ferranti, Laszlo Liptak, Karthik Nataraj

Theory & Applications of Graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal such sets are precisely sets of edges incident to a single vertex. The conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond these, and it is defined as the minimum number of edges whose deletion results in a graph with neither isolated vertices nor perfect matchings. In this paper we generalize this concept to get a hierarchy of …

A General Lower Bound On Gallai-Ramsey Numbers For Non-Bipartite Graphs, Colton Magnant

Theory & Applications of Graphs

Given a graph H and a positive integer k, the k-color Gallai-Ramsey number gr_k(K3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph K_n contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs H to a value that we conjecture to …

On The Planarity Of Generalized Line Graphs, Khawlah H. Alhulwah, Mohra Zayed, Ping Zhang

Theory & Applications of Graphs

One of the most familiar derived graphs is the line graph. The line graph L(G) of a graph G is that graph whose vertices are the edges of G where two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. Two nontrivial paths P and Q in a graph G are said to be adjacent paths in G if P and Q have exactly one vertex in common and this vertex is an end-vertex of both P and Q. For an integer ℓ ≥ 2, the ℓ -line graph Lℓ (G) of a …

A Survey On Monochromatic Connections Of Graphs, Xueliang Li, Di Wu

Theory & Applications of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it.
In this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.

Characterizations Of Some Classes Of Graphs That Are Nearly Series-Parallel, Victoria Fontaine

LSU Doctoral Dissertations

A series-parallel graph can be built from a single-edge graph by a sequence of series and parallel extensions. The class of such graphs coincides with the class of graphs that do not have the complete graph K₄ as a minor. This dissertation considers a class M₁ of graphs that are close to being series-parallel. In particular, every member of the class has the property that one can obtain a series-parallel graph by adding a new edge and contracting it out, or by splitting a vertex into two vertices whose neighbor sets partition the neighbor set of the original …

An Exploration Of The Chromatic Polynomial, Amanda Aydelotte

Mathematics Undergraduate Theses

In 1912, George Birkhoff was studying the Four Color Problem, and in doing so introduced the concept of the chromatic polynomial. While this did not end up directly contributing to proving that every map could be colored with four colors such that no region shares a border with another region of the same color, the chromatic polynomial has been found to have some very interesting properties. In this paper, it will be our goal to examine some of these properties and use them to determine information about their corresponding graphs.

Optimal Layout For A Component Grid, Michael W. Ebert

Computer Science and Software Engineering

Several puzzle games include a specific type of optimization problem: given components that produce and consume different resources and a grid of squares, find the optimal way to place the components to maximize output. I developed a method to evaluate potential solutions quickly and automated the solving of the problem using a genetic algorithm.

Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

Mathematics and Statistics Faculty Publications and Presentations

Consider a surface S and let M ⊂ S. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus g = 2 and g = 3. The classification for genus 0 and 1 was done …

Edge Colorings Of Complete Multipartite Graphs Forbidding Rainbow Cycles, Peter Johnson, Andrew Owens

Theory & Applications of Graphs

It is well known that if the edges of a finite simple connected graph on n vertices are colored so that no cycle is rainbow, then no more than n-1 colors can appear on the edges. In previous work it has been shown that the essentially different rainbow-cycle-forbidding edge colorings of K_n with n-1 colors appearing are in 1-1 correspondence with (can be encoded by) the (isomorphism classes of) full binary trees with n leafs. In the encoding, the natural Huffman labeling of each tree arising from the assignment of 1 to each leaf plays a role. Very recently …

A High Quality, Eulerian 3d Fluid Solver In C++, Lejon Anthony Mcgowan

Computer Science and Software Engineering

Fluids are a part of everyday life, yet are one of the hardest elements to properly render in computer graphics. Water is the most obvious entity when thinking of what a fluid simulation can achieve (and it is indeed the focus of this project), but many other aspects of nature, like fog, clouds, and particle effects. Real-time graphics like video games employ many heuristics to approximate these effects, but large-scale renderers aim to simulate these effects as closely as possible.

In this project, I wish to achieve effects of the latter nature. Using the Eulerian technique of discrete grids, I …

Graph Structures In Bipolar Neutrosophic Environment, Florentin Smarandache, Muhammad Akram, Muzzamal Sitara

Branch Mathematics and Statistics Faculty and Staff Publications

A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations.

An Extended Formulation Of The Convex Recoloring Problem On A Phylogenetic Tree, Sangho Shim

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Repeat And Return Patterns In Double Occurrence Words, Lukas Nabergall

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Some Results In Combinatorial Number Theory, Karl Levy

Dissertations, Theses, and Capstone Projects

The first chapter establishes results concerning equidistributed sequences of numbers. For a given $d\in\mathbb{N}$, $s(d)$ is the largest $N\in\mathbb{N}$ for which there is an $N$-regular sequence with $d$ irregularities. We compute lower bounds for $s(d)$ for $d\leq 10000$ and then demonstrate lower and upper bounds $\left\lfloor\sqrt{4d+895}+1\right\rfloor\leq s(d)< 24801d^{3} + 942d^{2} + 3$ for all $d\geq 1$. In the second chapter we ask if $Q(x)\in\mathbb{R}[x]$ is a degree $d$ polynomial such that for $x\in[x_k]=\{x_1,\cdots,x_k\}$ we have $|Q(x)|\leq 1$, then how big can its lead coefficient be? We prove that there is a unique polynomial, which we call $L_{d,[x_k]}(x)$, with maximum lead coefficient under these constraints and construct an algorithm that generates $L_{d,[x_k]}(x)$.

A Weighted Möbius Function, Derek Garton

Mathematics and Statistics Faculty Publications and Presentations

Fix an odd prime ℓ and let G be the poset of isomorphism classes of finite abelian ℓ-groups, ordered by inclusion. If ξ:G→R^≥0 is a discrete probability distribution on G and A ∈ G, define the Ath moment of ξ to be . The question of determining conditions that ensure ξ is completely determined by its moments has been of recent interest in many problems of Cohen–Lenstra type. Furthermore, recovering ξ from its moments requires a new Möbius-type inversion formula on G. In this paper, we define this function, relate it to the classical Möbius …

Ns-K-Nn: Neutrosophic Set-Based K-Nearest Neighbors Classifier, Florentin Smarandache, Yaman Akbulut, Abdulkadir Sengur, Yanhui Guo

Branch Mathematics and Statistics Faculty and Staff Publications

k-nearest neighbors (k-NN), which is known to be a simple and efficient approach, is a non-parametric supervised classifier. It aims to determine the class label of an unknown sample by its k-nearest neighbors that are stored in a training set. The k-nearest neighbors are determined based on some distance functions. Although k-NN produces successful results, there have been some extensions for improving its precision. The neutrosophic set (NS) defines three memberships namely T, I and F. T, I, and F shows the truth membership degree, the false membership degree, and the indeterminacy membership degree, respectively. In this paper, the NS …

Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log₂(α)³ + 16.5log₂(α)² + …

Antichains And Diameters Of Set Systems, Brent Mckain

Department of Mathematics: Dissertations, Theses, and Student Research

In this thesis, we present a number of results, mostly concerning set systems that are antichains and/or have bounded diameter. Chapter 1 gives a more detailed outline of the thesis. In Chapter 2, we give a new short proof of Kleitman's theorem concerning the maximal size of a set system with bounded diameter. In Chapter 3, we turn our attention to antichains with bounded diameter. Šileikis conjectured that an antichain of diameter D has size at most (n/D/2). We present several partial results towards the conjecture.

In 2014, Leader and Long gave asymptotic bounds on the size of …

Vertex Weighted Spectral Clustering, Mohammad Masum

Electronic Theses and Dissertations

Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to …

A Bijection On Classes Enumerated By The Schröder Numbers, Michael W. Schroeder, Rebecca Smith

Mathematics Faculty Research

We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine is known to be enumerated by the Schroder numbers. In this paper, we give a bijection between these sortable permutations of length n and Schroder paths of order n − 1: the lattice paths from (0, 0) to (n − 1, n − 1) composed of East steps (1, 0), North steps (0, 1), and Diagonal steps (1, …