Discrete Mathematics and Combinatorics Commons^™

Bounds For The Zero Forcing Number Of Graphs With Large Girth, Randy Davila, Franklin Kenter

Theory & Applications of Graphs

The zero-forcing number, Ζ(G) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Ζ(G) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ Ζ(G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Ζ(G) as a …

Scheduling N Burgers For A K-Burger Grill: Chromatic Numbers With Restrictions, Peter Johnson, Xiaoya Zha

Theory & Applications of Graphs

The chromatic number has a well-known interpretation in the area of scheduling. If the vertices of a finite, simple graph are committees, and adjacency of two committees indicates that they must never be in session simultaneously, then the chromatic number of the graph is the smallest number of hours during which the committees/vertices of the graph may all have properly scheduled meetings of one continuous hour each. Slivnik [3] showed that the fractional chromatic number can be similarly characterized. In that characterization, the meetings are allowed to be broken into a finite number of disjoint intervals. Here we consider chromatic …

Path-Tables Of Trees: A Survey And Some New Results, Kevin Asciak

Theory & Applications of Graphs

The (vertex) path-table of a tree Τ contains quantitative information about the paths in Τ. The entry (i,j) of this table gives the number of paths of length j passing through vertex ^vi. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree Τ does not determine Τ uniquely. We shall show that in trees, the number of paths passing through …

Connection And Separation In Hypergraphs, Mohammad A. Bahmanian, Mateja Sajna

Theory & Applications of Graphs

In this paper we study various fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. We prove a number of new results involving these concepts. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.

Community Detection Detailed For Online Social Networks, Christopher J. Hogan

Theses and Dissertations (Comprehensive)

Ever since the internet became publicly available it has allowed users to interact with each other across virtual networks. With this large amounts of data being collected the clustering of this information has become an even more powerful tool for recognize patterns and trends in a network. In this research we look build a model for Community Detection in these online social networks. We combine the ideas from both discrete mathematics and sociology, to build an algorithm with the specific intent on discovering communities that exist in an online social network. We present many of the sociology theories behind the …

Extremal Theorems For Degree Sequence Packing And The Two-Color Discrete Tomography Problem, Jennifer Diemunsch, Michael Ferrara, Sogol Jahanbekam, James Shook

Faculty Publications

No abstract provided.

Lucky Choice Number Of Planar Graphs With Given Girth, Axel Brandt, Jennifer Diemunsch, Sogol Jahanbekam

Faculty Publications

No abstract provided.

I Don't Play Chess: A Study Of Chess Piece Generating Polynomials, Stephen R. Skoch

Senior Independent Study Theses

This independent study examines counting problems of non-attacking rook, and non-attacking bishop placements. We examine boards for rook and bishop placement with restricted positions and varied dimensions. In this investigation, we discuss the general formula of a generating function for unrestricted, square bishop boards that relies on the Stirling numbers of the second kind. We discuss the maximum number of bishops we can place on a rectangular board, as well as a brief investigation of non-attacking rook placements on three-dimensional boards, drawing a connection to latin squares.

Domination Numbers Of Semi-Strong Products Of Graphs, Stephen R. Cheney

Theses and Dissertations

This thesis examines the domination number of the semi-strong product of two graphs G and H where both G and H are simple and connected graphs. The product has an edge set that is the union of the edge set of the direct product of G and H together with the cardinality of V(H), copies of G. Unlike the other more common products (Cartesian, direct and strong), the semi-strong product is neither commutative nor associative.

The semi-strong product is not supermultiplicative, so it does not satisfy a Vizing like conjecture. It is also not submultiplicative so it shares these two …

Graphs Of Classroom Networks, Rebecca Holliday

College of Graduate Studies: Theses & Dissertations

In this work, we use the Havel-Hakimi algorithm to visualize data collected from students to investigate classroom networks. The Havel-Hakimi algorithm uses a recursive method to create a simple graph from a graphical degree sequence. In this case, the degree sequence is a representation of the students in a classroom, and we use the number of peers with whom a student studied or collaborated to determine the degree of each. We expand upon the Havel-Hakimi algorithm by coding a program in MATLAB that generates random graphs with the same degree sequence. Then, we run another algorithm to find the isomorphism …

Labeled Trees And Spanning Trees: Computational Discrete Mathematics And Applications, Demet Yalman

College of Graduate Studies: Theses & Dissertations

In this thesis, we examine two topics. In the first part, we consider Leech tree which is a tree of order n with positive integer edge weights such that the weighted distances between pairs of vertices are exactly from 1 to n choose 2. Only five Leech trees are known and some non-existence results have been presented through the years. Variations of Leech trees such as the minimal distinct distance trees and modular Leech trees have been considered in recent years. In this thesis, such Leech-type questions on distances between leaves are studied as well as some other labeling questions …

Combinatorial Game Theory: An Introduction To Tree Topplers, John S. Ryals Jr.

College of Graduate Studies: Theses & Dissertations

The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing.

Quaternionic Hardy Spaces In The Open Unit Ball And Half Space And Blaschke Products, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Hardy spaces H2(B) and H2(H+), where B and H+ denote, respectively, the open unit ball of the quaternions and the half space of quaternions with positive real part, as well as Blaschke products, have been intensively studied in a series of papers where they are used as a tool to prove other results in Schur analysis. This paper gives an overview on the topic, collecting the various results available.

Self-Mappings Of The Quaternionic Unit Ball: Multiplier Properties, Schwarz-Pick Inequality, And Nevanlinna-Pick Interpolation Problem, Daniel Alpay, Vladimir Bolotnikov, Fabrizio Colombo, Irene Sabadini, Fabrizio Colombo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H2(B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

An Extension Of Herglotz's Theorem To The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only there exists a Cs×s-valued positive measure dμ on [0,2π] such that r(n)=∫2π0eintdμ(t)for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.

A New Resolvent Equation For The S-Functional Calculus, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is …

Infinite Product Representations For Kernels And Iteration Of Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping R in one complex variable, and its iterations.

On Algebras Which Are Inductive Limits Of Banach Spaces, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebra, and in particular, in the associated Wiener algebra.

Wiener-Chaos Approach To Optimal Prediction, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this work we combine Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on part of its sample path. This is done by considering special bases for the Gaussian space G generated by the process, which allows us to obtain an orthogonal basis for the Fock space of G such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample …

Coloring The Square Of Planar Graphs Without 4-Cycles Or 5-Cycles, Robert Jaeger

Theses and Dissertations

The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring.

To prove a stronger upper bound, we consider only planar graphs …