Planar Graphs, Biplanar Graphs And Graph Thickness, 2016 California State University - San Bernardino

#### Planar Graphs, Biplanar Graphs And Graph Thickness, Sean M. Hearon

*Electronic Theses, Projects, and Dissertations*

A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.

Potentially Nilpotent Tridiagonal Sign Patterns Of Order 4, 2016 North University of China

#### Potentially Nilpotent Tridiagonal Sign Patterns Of Order 4, Yubin Gao, Yanling Shao

*Electronic Journal of Linear Algebra*

An $n\times n$ sign pattern ${\cal A}$ is said to be potentially nilpotent (PN) if there exists some nilpotent real matrix $B$ with sign pattern ${\cal A}$. In [M.~Arav, F.~Hall, Z.~Li, K.~Kaphle, and N.~Manzagol.Spectrally arbitrary tree sign patterns of order $4$, {\em Electronic Journal of Linear Algebra}, 20:180--197, 2010.], the authors gave some open questions, and one of them is the following: {\em For the class of $4 \times 4$ tridiagonal sign patterns, is PN (together with positive and negative diagonal entries) equivalent to being SAP?}\ In this paper, a positive answer ...

Bounds On The Expected Size Of The Maximum Agreement Subtree, 2016 The Ohio State University

#### Bounds On The Expected Size Of The Maximum Agreement Subtree, Colby Long, Daniel Irving Bernstein, Lam Si Tung Ho, Mike Steel, Katherine St. John, Seth Sullivant

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Exploring The Space Of Rna Secondary Structures, 2016 Georgia Institute of Technology

#### Exploring The Space Of Rna Secondary Structures, Heather C. Smith

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Markov Chains On Graphical Models Of Gene Regulation, 2016 Georgia Institute of Technology

#### Markov Chains On Graphical Models Of Gene Regulation, Megan Bernstein

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Genome Rearrangement: Graphs And Matrices, 2016 Emory University

#### Genome Rearrangement: Graphs And Matrices, Jeffrey Davis

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

On The Expected Number Of Crossings In A Tanglegram, 2016 University of South Carolina

#### On The Expected Number Of Crossings In A Tanglegram, Eva Czabarka

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

On The Perfect Reconstruction Of The Structure Of Dynamic Networks, 2016 University of Dayton

#### On The Perfect Reconstruction Of The Structure Of Dynamic Networks, Alan Veliz-Cuba

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Spectral Graph Theory And New Stability Measures For Deterministic Gene Regulatory Networks, 2016 Illinois State University

#### Spectral Graph Theory And New Stability Measures For Deterministic Gene Regulatory Networks, Fusun Akman, Devin Akman

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Rogers-Ramanujan Computer Searches, 2016 West Chester University of Pennsylvania

#### Rogers-Ramanujan Computer Searches, James Mclaughlin, Andrew Sills, Peter Zimmer

*James McLaughlin*

We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers–Ramanujan type and identities of false theta functions.

Preservers Of Term Ranks And Star Cover Numbers Of Symmetric Matrices, 2016 Utah State University

#### Preservers Of Term Ranks And Star Cover Numbers Of Symmetric Matrices, Leroy B. Beasley

*Electronic Journal of Linear Algebra*

Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of ...

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 ...

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 ...

Clustering-Based Robot Navigation And Control, 2016 University of Pennsylvania

#### Clustering-Based Robot Navigation And Control, Omur Arslan

*Departmental Papers (ESE)*

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths, capturing the connectivity of the underlying space. This dissertation introduces the use of clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in ...

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 ...