On The Inverse Of A Class Of Weighted Graphs, 2018 Indian Statistical Institute - Delhi Center

#### On The Inverse Of A Class Of Weighted Graphs, Swarup Kumar Panda, Sukanta Pati

*Electronic Journal of Linear Algebra*

In this article, only connected bipartite graphs $G$ with a unique perfect matching $\c{M}$ are considered. Let $G_\w$ denote the weighted graph obtained from $G$ by giving weights to its edges using the positive weight function $\w:E(G)\ar (0,\ity)$ such that $\w(e)=1$ for each $e\in\c{M}$. An unweighted graph $G$ may be viewed as a weighted graph with the weight function $\w\equiv\1$ (all ones vector). A weighted graph $G_\w$ is nonsingular if its adjacency matrix $A(G_\w)$ is nonsingular. The {\em inverse} of a nonsingular weighted graph ...

On The Second Least Distance Eigenvalue Of A Graph, 2018 College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, P.R.China.

#### On The Second Least Distance Eigenvalue Of A Graph, Xueyi Huang, Qiongxiang Huang, Lu Lu

*Electronic Journal of Linear Algebra*

Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\partial_1(G)\ge\partial_2(G)\ge\cdots\ge\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, the connected graphs with @n1(G) at least the smallest root of $x^3=3x^2-11x-6 = 0$ are determined. Additionally, some non-isomorphic distance cospectral graphs are given.

Traveling In Networks With Blinking Nodes, 2018 Southern CT State University

#### Traveling In Networks With Blinking Nodes, Braxton Carrigan, James Hammer

*Theory and Applications of Graphs*

We say that a blinking node system modulo $n$ is an ordered pair $(G,L)$ where $G$ is a graph and $L$ is an on-labelling which indicates when vertices can be visited. An On-Hamiltonian walk is a sequence of all the vertices of $G$ such that the position of each vertex modulo $n$ is an integer of the label of that vertex. This paper will primarily investigate finding the shortest On-Hamiltonian walks in a blinking node system on complete graphs and complete bipartite graphs but also establishes the terminology and initial observations for working with blinking node systems on other ...

The Graphs And Matroids Whose Only Odd Circuits Are Small, 2018 Louisiana State University and Agricultural and Mechanical College

#### The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler

*LSU Doctoral Dissertations*

This thesis is motivated by a graph-theoretical result of Maffray, which states that a 2-connected graph with no odd cycles exceeding length 3 is bipartite, is isomorphic to K_4, or is a collection of triangles glued together along a common edge. We first prove that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is M(K_4) or F_7, or M is the cycle matroid of a graph consisting of a collection of triangles glued together along a common edge. This result implies that a 2-connected loopless graph ...

On The Density Of The Odd Values Of The Partition Function, 2018 Michigan Technological University

#### 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 ...

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, 2018 Murray State University

#### 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 associated with ...

On The Domination Chain Of M By N Chess Graphs, 2018 Murray State University

#### 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.

A Combinatorial Miscellany: Antipodes, Parking Cars, And Descent Set Powers, 2018 University of Kentucky

#### A Combinatorial Miscellany: Antipodes, Parking Cars, And Descent Set Powers, Alexander Thomas Happ

*Theses and Dissertations--Mathematics*

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results ...

A General Lower Bound On Gallai-Ramsey Numbers For Non-Bipartite Graphs, 2018 Georgia Southern University

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

*Theory and Applications of Graphs*

Given a graph $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number $gr_{k}(K_{3} : H)$ 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 be sharp ...

Pbw Bases And Marginally Large Tableaux In Type D, 2018 Central Michigan University

#### 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

A Survey On Monochromatic Connections Of Graphs, 2018 Nankai University

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

*Theory and 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.

Hilbert Bases, Descent Statistics, And Combinatorial Semigroup Algebras, 2018 University of Kentucky

#### Hilbert Bases, Descent Statistics, And Combinatorial Semigroup Algebras, Mccabe J. Olsen

*Theses and Dissertations--Mathematics*

The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials.

Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory.

In Chapter ...

Polytopes Associated To Graph Laplacians, 2018 University of Kentucky

#### Polytopes Associated To Graph Laplacians, Marie Meyer

*Theses and Dissertations--Mathematics*

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the *Laplacian simplex*, *P _{G}*, obtained from a finite graph

*G*by taking the convex hull of the columns of the Laplacian matrix for

*G*. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph

*D*to obtain the

*Laplacian polytope*,

*P*.

_{D}Basic properties of both families of simplices, *P _{G}* and

*P*, are established using techniques ...

_{D}Fine Structure Of 4-Critical Triangle-Free Graphs Iii. General Surfaces, 2018 Charles University, Prague

#### Fine Structure Of 4-Critical Triangle-Free Graphs Iii. General Surfaces, Zdenek Dvorak, Bernard Lidicky

*Mathematics Publications*

Dvořák, Král', and Thomas [*Three-Coloring Triangle-Free Graphs on Surfaces* IV. *Bounding Face Sizes of 4-Critical Graphs*, preprint, arXiv:1404.6356v3, 2015; *Three-Coloring Triangle-Free Graphs on Surfaces* VI. *3-Colorability of Quadrangulations*, preprint, arXiv:1509.01013, 2015] gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description, however, contains two substructures (both related to graphs embedded in a plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.

3-Maps And Their Generalizations, 2018 Virginia Commonwealth University

#### 3-Maps And Their Generalizations, Kevin J. Mccall

*Theses and Dissertations*

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

Uses Of Mathematics In Computer Animation And 3d Rendering Software, 2018 University of Colorado, Boulder

#### Uses Of Mathematics In Computer Animation And 3d Rendering Software, Peter Rock

*Undergraduate Honors Theses*

As the title suggests, this paper discusses the applications of several mathematical concepts to computer animation software generally used in the creation of movies and video games. Topics covered will include differential forms, conformal maps, surface texturing, and lighting techniques. It is not the goal of this paper to present anything particularly novel to the mathematical community, but rather to present something that is entertaining to read that will hopefully engage both mathematicians and sane people alike. This paper has been carefully crafted so that it should be accessible to most people with a Calc. I background. That being said ...

Extensions Of The Morse-Hedlund Theorem, 2018 Bucknell University

#### Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

*Honors Theses*

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...*ababababab*...". The Morse-Hedlund theorem says that a bi-infinite word *f* repeats itself, in at most *n* letters, if and only if the number of distinct subwords of length *n* is at most *n*. Using the example, "...*ababababab*...", there are 2 subwords of length 3, namely "*aba*" and "*bab*". Since 2 is less than 3, we must have that "...*ababababab*..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. Interestingly, there are many extensions of this theorem to multiple dimensions ...

Spn Graphs And Rank-1 Cp-Completable Graphs, 2018 Iowa State University

#### Spn Graphs And Rank-1 Cp-Completable Graphs, Leslie Hogben, Naomi Shaked-Monderer

*Mathematics Publications*

A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [Shaked-Monderer, SPN graphs: When copositive = SPN, *Linear Algebra Appl.*, 509(15):82-113, 2016], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. We disprove this conjecture, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K4 is a SPN graph if and only if at most one edge is subdivided. We conjecture that a graph is an SPN graph ...

Zero Forcing And Power Domination For Graph Products, 2018 Westminster College - Fulton

#### Zero Forcing And Power Domination For Graph Products, Katherine F. Benson, Daniela Ferrero, Mary Flagg, Veronika Furst, Leslie Hogben, Violeta Vasilevska, Brian Wissman

*Mathematics Publications*

The power domination number arose from the monitoring of electrical networks, and methods for its determination have the associated application. The zero forcing number arose in the study of maximum nullity among symmetric matrices described by a graph (and also in control of quantum systems and in graph search algorithms). There has been considerable effort devoted to the determination of the power domination number, the zero forcing number, and maximum nullity for specific families of graphs. In this paper we exploit the natural relationship between power domination and zero forcing to obtain results for the power domination number of tensor ...

Polychromatic Colorings On The Hypercube, 2018 West Virginia University

#### Polychromatic Colorings On The Hypercube, John Goldwasser, Bernard Lidicky, Ryan R. Martin, David Offner, John Talbot, Michael Young

*Mathematics Publications*

Given a subgraph G of the hypercube Qn, a coloring of the edges of Qn such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a specific type of coloring, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of ...