Nested (2,R)-Regular Graphs And Their Network Properties., 2012 East Tennessee State University

#### Nested (2,R)-Regular Graphs And Their Network Properties., Josh Daniel Brooks

*Electronic Theses and Dissertations*

A graph *G* is a (*t*, *r*)-regular graph if every collection of *t* independent vertices is collectively adjacent to exactly *r* vertices. If a graph *G* is (2, *r*)-regular where *p*, *s*, and *m* are positive integers, and *m* ≥ 2, then when *n* is sufficiently large, then *G* is isomorphic to *G = K _{s}+mK_{p}*, where 2(

*p*-1)+

*s*=

*r*. A nested (2,

*r*)-regular graph is constructed by replacing selected cliques with a (2,

*r*)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested '

*s*' graph when

*n = s ...*

Global Domination Stable Graphs, 2012 East Tennessee State University

#### Global Domination Stable Graphs, Elizabeth Marie Harris

*Electronic Theses and Dissertations*

A set of vertices *S* in a graph *G* is a global dominating set (GDS) of *G* if *S* is a dominating set for both *G* and its complement *G*. The minimum cardinality of a global dominating set of *G* is the global domination number of *G*. We explore the effects of graph modifications on the global domination number. In particular, we explore edge removal, edge addition, and vertex removal.

A Survey Of Classical And Recent Results In Bin Packing Problem, 2012 University of Nevada, Las Vegas

#### A Survey Of Classical And Recent Results In Bin Packing Problem, Yoga Jaideep Darapuneni

*UNLV Theses, Dissertations, Professional Papers, and Capstones*

In the classical bin packing problem one receives a sequence of n items 1, 2,..., n with sizes s1, s2, . . . ,sn where each item has a fixed size in (0, 1]. One needs to find a partition of the items into sets of size1, called bins, so that the number of sets in the partition is minimized and the sum of the sizes of the pieces assigned to any bin does not exceed its capacity. This combinatorial optimization problem which is NP hard has many variants as well as online and offline versions of the problem. Though the problem is ...

Bipartizing Fullerenes, 2012 Charles University

#### Bipartizing Fullerenes, Zdeněk Dvořák, Bernard Lidicky, Riste Škrekovskib

*Bernard Lidický*

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, 2012 University of Nebraska-Lincoln

#### The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

*Dissertations, Theses, and Student Research Papers in Mathematics*

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might ...

Parameters Related To Tree-Width, Zero Forcing, And Maximum Nullity Of A Graph, 2012 University of Tennessee, Chattanooga

#### Parameters Related To Tree-Width, Zero Forcing, And Maximum Nullity Of A Graph, Francesco Barioli, Wayne Barrett, Shaun M. Fallat, Leslie Hogben, Bryan Shader, P. Van Den Driessche, Hein Van Der Holst

*Mathematics Publications*

Tree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d'arborescence, path-width, and proper ...

Vertex And Edge Spread Of Zero Forcing Number, Maximum Nullity, And Minimum Rank Of A Graph, 2012 Willamette University

#### Vertex And Edge Spread Of Zero Forcing Number, Maximum Nullity, And Minimum Rank Of A Graph, Christina J. Edholm, Leslie Hogben, My Huynh, Josh Lagrange, Darren D. Row

*Mathematics Publications*

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i not equal j) is nonzero whenever {i, j} is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on ...

On The Graph Complement Conjecture For Minimum Rank, 2012 University of Tennessee, Chattanooga

#### On The Graph Complement Conjecture For Minimum Rank, Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Hein Van Der Holst

*Mathematics Publications*

The minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus–Gaddum type problem involving the graph parameter minimum rank. The conjectured bound is the order of the graph plus two. Other variants ...

A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For, 2012 Smith College

#### A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For, Julianna Tymoczko

*Mathematics and Statistics: Faculty Publications*

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and xxx_{2}. Building on a result of Kuperberg’s, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3 × n and irreducible webs for xxx_{3} whose boundary vertices are all sources. In this paper, we give a simple and explicit map from standard Young tableaux of shape 3 × n ...

Liar's Domination In Grid Graphs, 2012 East Tennessee State University

#### Liar's Domination In Grid Graphs, Christopher Kent Sterling

*Electronic Theses and Dissertations*

As introduced by Slater in 2008, liar's domination provides a way of modeling protection devices where one may be faulty. Assume each vertex of a graph *G* is the possible location for an intruder such as a thief. A protection device at a vertex *v* is assumed to be able to detect the intruder at any vertex in its closed neighborhood N[*v*] and identify at which vertex in N[*v*] the intruder is located. A dominating set is required to identify any intruder's location in the graph *G*, and if any one device can fail to detect ...

Preferential Arrangement Containment In Strict Superpatterns, 2012 East Tennessee State University

#### Preferential Arrangement Containment In Strict Superpatterns, Martha Louise Liendo

*Electronic Theses and Dissertations*

Most results on pattern containment deal more directly with pattern avoidance, or the enumeration and characterization of strings which avoid a given set of patterns. Little research has been conducted regarding the word size required for a word to contain all patterns of a given set of patterns. The set of patterns for which containment is sought in this thesis is the set of preferential arrangements of a given length. The term preferential arrangement denotes strings of characters in which repeated characters are allowed, but not necessary. Cardinalities for sets of all preferential arrangements of given lengths and alphabet sizes ...

The Rook-Brauer Algebra, 2012 Macalester College

#### The Rook-Brauer Algebra, Elise G. Delmas

*Mathematics, Statistics, and Computer Science Honors Projects*

We introduce an associative algebra RB_{k}(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2*k* vertices. The rook-Brauer algebra contains the group algebra of the symmetric group, the Brauer algebra, and the rook monoid algebra as subalgebras. We show that the basis of RB_{k}(x) is generated by special diagrams s_{i}, t_{i} (1 <= i < *k*) and p_{j} (1 <= j <= *k*), where the s_{i} are the simple transpositions that generated the symmetric group S_{k}, the t_{i} are the "contraction maps" which generate the Brauer algebra B_{k ...}

On The Number Of Tilings Of A Square By Rectangles, 2012 University of Tennessee - Knoxville

#### On The Number Of Tilings Of A Square By Rectangles, Timothy Michaels

*Chancellor’s Honors Program Projects*

No abstract provided.

Generalized Branching In Circle Packing, 2012 University of Tennessee, Knoxville

#### Generalized Branching In Circle Packing, James Russell Ashe

*Doctoral Dissertations*

Circle packings are configurations of circle with prescribed patterns of tangency. They relate to a surprisingly diverse array of topics. Connections to Riemann surfaces, Apollonian packings, random walks, Brownian motion, and many other topics have been discovered. Of these none has garnered more interest than circle packings' relationship to analytical functions. With a high degree of faithfulness, maps between circle packings exhibit essentially the same geometric properties as seen in classical analytical functions. With this as motivation, an entire theory of discrete analytic function theory has been developed. However limitations in this theory due to the discreteness of circle packings ...

Extremal Problems For Roman Domination, 2012 Illinois Mathematics and Science Academy

#### Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West

*Noah Prince*

A *Roman dominating function *of a graph *G* is a labeling *f: V*(*G*) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The *Roman domination number **γ** _{R}*(

*G*) of

*G*is the minimum of ∑

_{ʋϵ}

_{V(}

_{G}_{)}

*f*(

*v*) over such functions. Let

*G*be a connected

*n*-vertex graph. We prove that

*γ*

*(*

_{R}*G*) ≤ 4

*n*/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for

*γ*

*(*

_{R}*G*) +

*γ*

_{R}(*Ḡ*

*)*and

*γ*

*(*

_{R}*G*)

*γ*

_{R}(*Ḡ*

*)*, improving known results ...

Combinatorics Using Computational Methods, 2012 University of Nebraska-Lincoln

#### Combinatorics Using Computational Methods, Derrick Stolee

*Dissertations, Theses, and Student Research Papers in Mathematics*

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.

The 1, 2-Conjecture For Graphs With Relatively Small Chromatic Number, 2012 University of Illinois at Urbana–Champaign

#### The 1, 2-Conjecture For Graphs With Relatively Small Chromatic Number, Sogol Jahanbekam, Douglas West

*Faculty Publications*

No abstract provided.

Exploring The On-Line Partitioning Of Posets Problem, 2012 Scripps College

#### Exploring The On-Line Partitioning Of Posets Problem, Leah F. Rosenbaum

*Scripps Senior Theses*

One question relating to partially ordered sets (posets) is that of partitioning or dividing the poset's elements into the fewest number of chains that span the poset. In 1950, Dilworth established that the width of the poset - the size of the largest set composed only of incomparable elements - is the minimum number of chains needed to partition that poset. Such a bound in on-line partitioning has been harder to establish, and work has evalutated classes of posets based on their width. This paper reviews the theorems that established val(2)=5 and illustrates them with examples. It also covers ...

Session D-3: Discrete Mathematics: A Great Curriculum Connector, 2012 Illinois Mathematics and Science Academy

#### Session D-3: Discrete Mathematics: A Great Curriculum Connector, Donald Porzio

*Professional Learning Day*

Many topics that fall under the umbrella of Discrete Mathematics cut across the traditional high school curriculum areas of algebra, geometry, and pre-calculus. Come try some classroom-ready hands-on Discrete Mathematics activities that illustrate the true interconnectedness of mathematics.

Fixed Points And Excedances In Restricted Permutations, 2012 Dartmouth College

#### Fixed Points And Excedances In Restricted Permutations, Sergi Elizalde

*Open Dartmouth: Faculty Open Access Articles*

Using an unprecedented technique involving diagonals of non-rational generating functions, we prove that among the permutations of length $n$ with $i$ fixed points and $j$ excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for any given $i,j$. Our theorem generalizes a result of Robertson, Saracino and Zeilberger. Even though bijective proofs have later been found by the author jointly with Pak and with Deutsch, this paper contains the original analytic proof that was presented at FPSAC 2003.