Induced Subgraph Saturated Graphs, 2016 University of New England

#### Induced Subgraph Saturated Graphs, Craig M. Tennenhouse

*Theory and Applications of Graphs*

A graph $G$ is said to be \emph{$H$-saturated} if $G$ contains no subgraph isomorphic to $H$ but the addition of any edge between non-adjacent vertices in $G$ creates one. While induced subgraphs are often studied in the extremal case with regard to the removal of edges, we extend saturation to induced subgraphs. We say that $G$ is \emph{induced $H$-saturated} if $G$ contains no induced subgraph isomorphic to $H$ and the addition of any edge to $G$ results in an induced copy of $H$. We demonstrate constructively that there are non-trivial examples of saturated graphs for all ...

An Extremal Problem For Finite Lattices, 2016 West Virginia University

#### An Extremal Problem For Finite Lattices, John Goldwasser, Brendan Nagle, Andres Saez

*John Copeland Nagle*

For a fixed M x N integer lattice L(M,N), we consider the maximum size of a subset A of L(M,N) which contains no squares of prescribed side lengths k(1),...,k(t). We denote this size by ex(L(M,N), {k(1),...,k(t)}), and when t = 1, we abbreviate this parameter to ex(L(M,N), k), where k = k(1). Our first result gives an exact formula for ex(L(M,N), k) for all positive integers k, M, and N, where ex(L(M,N), k) = ((3/4) + o(1)) MN holds ...

Improving Proof-Writing With Reading Guides, 2016 Dordt College

#### Improving Proof-Writing With Reading Guides, Mike Janssen

*Faculty Work: Comprehensive List*

One of the barriers in the transition to advanced mathematics is that the proofs and ideas in even the best mathematics texts must be read more carefully than many students are accustomed to. Yet in order to learn to write proofs well, one must learn how to read proofs well. Borrowing an idea from Lewis Ludwig, I flipped my introduction to proofs (discrete structures) course in Spring 2016 with the use of reading guides. Each day, students were responsible for reading a section of the text and completing a worksheet that highlighted the main points, asked students to create their ...

Potentially Eventually Positive Star Sign Patterns, 2016 Huaiyin Institute of Technology

#### Potentially Eventually Positive Star Sign Patterns, Yu Ber-Lin, Huang Ting-Zhu, Jie Cui, Deng Chunhua

*Electronic Journal of Linear Algebra*

An $n$-by-$n$ real matrix $A$ is eventually positive if there exists a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is potentially eventually positive (PEP) if there exists an eventually positive real matrix $A$ with the same sign pattern as $\mathcal{A}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is a minimal potentially eventually positive sign pattern (MPEP sign pattern) if $\mathcal{A}$ is PEP and no proper subpattern of $\mathcal{A}$ is PEP. Berman, Catral, Dealba, et al. [Sign patterns that ...

Combinatorics Of The Sonnet, 2016 The Open University

#### Combinatorics Of The Sonnet, Terry S. Griggs

*Journal of Humanistic Mathematics*

Using a definition of a sonnet, the number of basic rhyming schemes is enumerated. This is then used to discuss the 86 sonnets which appear in John Clare's *The Rural Muse*.

Hybrid Proofs Of The Q-Binomial Theorem And Other Identities, 2016 University of California - Irvine

#### Hybrid Proofs Of The Q-Binomial Theorem And Other Identities, Dennis Eichhorn, James Mclaughlin, Andrew V. Sills

*James McLaughlin*

No abstract provided.

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, 2016 West Chester University of Pennsylvania

#### Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew V. Sills

*James McLaughlin*

No abstract provided.

A New Summation Formula For Wp-Bailey Pairs, 2016 West Chester University of Pennsylvania

#### A New Summation Formula For Wp-Bailey Pairs, James Mclaughlin

*James McLaughlin*

No abstract provided.

Graphs With Reciprocal Eigenvalue Properties, 2016 IIT GUWAHATI

#### Graphs With Reciprocal Eigenvalue Properties, Swarup Kumar Panda, Sukanta Pati

*Electronic Journal of Linear Algebra*

In this paper, only simple graphs are considered. A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. From the definitions property SR implies property R. Furthermore, for some classes of graphs (for ...

Kemeny's Constant And An Analogue Of Braess' Paradox For Trees, 2016 University of Manitoba

#### Kemeny's Constant And An Analogue Of Braess' Paradox For Trees, Steve Kirkland, Ze Zeng

*Electronic Journal of Linear Algebra*

Given an irreducible stochastic matrix M, Kemeny’s constant K(M) measures the expected time for the corresponding Markov chain to transition from any given initial state to a randomly chosen final state. A combinatorially based expression for K(M) is provided in terms of the weights of certain directed forests in a directed graph associated with M, yielding a particularly simple expression in the special case that M is the transition matrix for a random walk on a tree. An analogue of Braess’ paradox is investigated, whereby inserting an edge into an undirected graph can increase the value of ...

Classifying Resolving Lists By Distances Between Members, 2016 University of Texas of the Permian Basin

#### Classifying Resolving Lists By Distances Between Members, Paul Feit

*Theory and Applications of Graphs*

L

Let $G$ be a connected graph and let $w_1,\cdots w_r$ be a list of vertices. We refer the choice of a triple $(r;G;w_1,\cdots w_r)$, as a {\em metric selection.} Let $\rho$ be the shortest path metric of $G$. We say that $w_1,\cdots w_r$ {\em resolves $G$ (metricly)\/} if the function $c:V(G)\mapsto\bbz^r$ given by

\[ x\mapsto (\rho (w_1,x),\cdots ,\rho (w_r,x))\]

is injective. We refer to this function the {\em code map,} and to its image as the {\em codes\/} of the triple $(r;G;w_1,\cdots ,w_r ...

Group-Antimagic Labelings Of Multi-Cyclic Graphs, 2016 Illinois Wesleyan University

#### Group-Antimagic Labelings Of Multi-Cyclic Graphs, Dan Roberts, Richard M. Low

*Theory and Applications of Graphs*

Let $A$ be a non-trivial abelian group. A connected simple graph $G = (V, E)$ is $A$-\textbf{antimagic} if there exists an edge labeling $f: E(G) \to A \backslash \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \Sigma$ $\{f(u,v): (u, v) \in E(G) \}$, is a one-to-one map. The \textit{integer-antimagic spectrum} of a graph $G$ is the set IAM$(G) = \{k: G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic and } k \geq 2\}$. In this paper, we analyze the integer-antimagic spectra for various classes of multi-cyclic graphs.

An Expansion Property Of Boolean Linear Maps, 2016 Shanghai Jiao Tong University

#### An Expansion Property Of Boolean Linear Maps, Yaokun Wu, Zeying Xu, Yinfeng Zhu

*Electronic Journal of Linear Algebra*

Given a finite set $K$, a Boolean linear map on $K$ is a map $f$ from the set $2^K$ of all subsets of $K$ into itself with $f(\emptyset )=\emptyset$ such that $f(A\cup B)=f(A)\cup f(B)$ holds for all $A,B\in 2^K$. For fixed subsets $X, Y$ of $K$, to predict if $Y$ is reachable from $X$ in the dynamical system driven by $f$, one can assume the existence of nonnegative integers $h$ with $f^h(X)=Y$, find an upper bound $\alpha$ for the minimum of all such assumed integers $h ...

Chromatic Connectivity Of Graphs, 2016 Western Michigan University

Upset Paths And 2-Majority Tournaments, 2016 California State University, San Bernardino

#### Upset Paths And 2-Majority Tournaments, Rana Ali Alshaikh

*Electronic Theses, Projects, and Dissertations*

In 2005, Alon, et al. proved that tournaments arising from majority voting scenarios have minimum dominating sets that are bounded by a constant that depends only on the notion of what is meant by a majority. Moreover, they proved that when a majority means that Candidate A beats Candidate B when Candidate A is ranked above Candidate B by at least two out of three voters, the tournament used to model this voting scenario has a minimum dominating set of size at most three. This result gives 2-majority tournaments some significance among all tournaments and motivates us to investigate when ...

Ádám's Conjecture And Arc Reversal Problems, 2016 California State University - San Bernardino

#### Ádám's Conjecture And Arc Reversal Problems, Claudio D. Salas

*Electronic Theses, Projects, and Dissertations*

A. Ádám conjectured that for any non-acyclic digraph *D*, there exists an arc whose reversal reduces the total number of cycles in *D*. In this thesis we characterize and identify structure common to all digraphs for which Ádám's conjecture holds. We investigate quasi-acyclic digraphs and verify that Ádám's conjecture holds for such digraphs. We develop the notions of arc-cycle transversals and reversal sets to classify and quantify this structure. It is known that Ádám's conjecture does not hold for certain infinite families of digraphs. We provide constructions for such counterexamples to Ádám's conjecture. Finally, we address ...

Hill's Diagrammatic Method And Reduced Graph Powers, 2016 The College of William & Mary

#### Hill's Diagrammatic Method And Reduced Graph Powers, Gregory D. Smith, Richard Hammack

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Discrete Stability Of Dpg Methods, 2016 Portland State University

#### Discrete Stability Of Dpg Methods, Ammar Harb

*Dissertations and Theses*

This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is ...

The Distance Spectral Radius Of Graphs With Given Number Of Odd Vertices, 2016 South China Normal University

#### The Distance Spectral Radius Of Graphs With Given Number Of Odd Vertices, Hongying Lin, Bo Zhou

*Electronic Journal of Linear Algebra*

The graphs with smallest, respectively largest, distance spectral radius among the connected graphs, respectively trees with a given number of odd vertices, are determined. Also, the graphs with the largest distance spectral radius among the trees with a given number of vertices of degree 3, respectively given number of vertices of degree at least 3, are determined. Finally, the graphs with the second and third largest distance spectral radius among the trees with all odd vertices are determined.

Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, 2016 Portland State University

#### Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

*Student Research Symposium*

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.