Physical Sciences and Mathematics Commons^™

Rearrangement Operations On Unrooted Phylogenetic Networks, Remie Janssen, Jonathan Klawitter

Theory and Applications of Graphs

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including …

Conditional Strong Matching Preclusion Of The Alternating Group Graph, Mohamad Abdallah, Eddie Cheng

Theory and Applications of Graphs

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the n-dimensional alternating group graph AG_n.

Forcibly-Biconnected Graphical Degree Sequences: Decision Algorithms And Enumerative Results, Kai Wang

Theory and Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n. Based on these enumerative …

Laplacian Spectral Characterization Of Signed Sun Graphs, Fatemeh Motialah, Mohammad Hassan Shirdareh Haghighi

Theory and Applications of Graphs

A sun SG_n is a graph of order 2n consisting of a cycle C_n, n ≥ 3, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if n is odd.

Fractional Strong Matching Preclusion For Two Variants Of Hypercubes, Huifen Ge, Tianlong Ma, Miaolin Wu, Yuzhi Xiao

Theory and Applications of Graphs

Let F be a subset of edges and vertices of a graph G. If G-F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each.

Colored Complete Hypergraphs Containing No Rainbow Berge Triangles, Colton Magnant

Theory and Applications of Graphs

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs.

Maximum Oriented Forcing Number For Complete Graphs, Yair Caro, Ryan Pepper

Theory and Applications of Graphs

The maximum oriented k-forcing number of a simple graph G, written MOF_k(G), is the maximum directed k-forcing number among all orientations of G. This invariant was recently introduced by Caro, Davila and Pepper in [6], and in the current paper we study the special case where G is the complete graph with order n, denoted K_n . While MOF_k(G) is an invariant for the underlying simple graph G, MOF_k(K_n) can also be interpreted as an interesting property for tournaments. Our main results …

Matching Preclusion Of The Generalized Petersen Graph, Ajay Arora, Eddie Cheng, Christopher Melekian

Theory and Applications of Graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. In this paper we determine the matching preclusion number for the generalized Petersen graph P(n,k) and classify the optimal sets.

Series-Parallel Operations With Alpha-Graphs, Christian Barrientos, Sarah Minion

Theory and Applications of Graphs

Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path Ρ₁₁}, are α-graphs. We also show that the one-point union of up to four copies of Κ_n,n is an …

Fractional Matching Preclusion For Butterfly Derived Networks, Xia Wang, Tianlong Ma, Chengfu Ye, Yuzhi Xiao, Fang Wang

Theory and Applications of Graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [18] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G, denoted by fmp(G), is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G, denoted by fsmp(G), is the minimum number of vertices and edges whose deletion …

Singular Ramsey And Turán Numbers, Yair Caro, Zsolt Tuza

Theory and Applications of Graphs

We say that a subgraph F of a graph G is singular if the degrees d_G(v) are all equal or all distinct for the vertices v ∈ V (F). The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m at least n, in every edge 2-coloring of K_m, at least one of the color classes contains F as a singular subgraph. In a similar flavor, the singular Turán number Ts(n,F) is defined as the maximum number of edges in a graph of order n, …

Preparing Pre-Service Teachers To Present At A State Conference, Heidi Eisenreich

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

As mathematics teacher educators, we have a responsibility to prepare as many people as we can, to teach mathematics in ways that foster a deeper understanding of the content. We do this by teaching current and future teachers in college programs and providing professional development to in-service teachers. A less explored way is to prepare these “students” to present ideas they have learned to colleagues at their school, other schools in their district, and conferences. In this paper, I share my experience of helping students go through the process of preparing to present over the last two years at our …

K-2 Mathematicians & Writers: Professional Learning Communities For Developing Conceptual Understanding, Doris Santarone, Angel R. Abney, Sandra M. Webb

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

National Council of Teachers of Mathematics (NCTM) has long supported the use of children’s literature, writing, and manipulatives to improve conceptual understanding of mathematics (2000). In a professional learning community for K-2 teachers, professional development was designed and implemented on ways to incorporate literacy and manipulatives into a mathematics lesson. The teachers were charged with collaboratively planning lessons that included multiple components: the standard(s), a mathematics activity, manipulatives, a writing task, and children’s literature. As the data were analyzed, it became apparent that while most of the lessons were well connected, this did not happen for all of the lessons. …

Creative Writing In The Mathematics Classroom, William Lacefield, Laura Markert

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Creative writing in the mathematics classroom promotes mathematical applications in the real world, constructivist learning, embodied learning, transfer of mathematical ideas, and student engagement. When students are allowed to write about mathematical concepts creatively, they are able to take concepts that they have learned and put them into their world or even create a situation where the mathematical concept applies. Applying mathematical concepts to other environments helps learners transfer mathematical concepts. Learners are able to take the mathematics content and contextualize it outside of the classroom. Writing in mathematics also is a way for students to embody learning. Because writing …

Conceptualizing And Interpreting Mean And Median With Future Teachers, Eryn M. Stehr, Ha Nguyen, Gregory Chamblee, Sharon Taylor

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Mathematical Education of Teachers II (METII), echoed by the American Statistical Association publication, Statistical Education of Teachers, recommended teacher preparation programs support future teachers in developing deep understandings of mean and median, such that middle grades teachers may use them to “summarize, describe, and compare distributions” (Conference Board of Mathematical Sciences, 2012, p. 44; Franklin et al., 2015). Georgia Standards of Excellence require statistical reasoning from students beginning as early as 6-7 years old, including interpretation of measures of center and statistical reasoning about best measures of center (Georgia Department of Education, 2015). This level of understanding and interpretation of …

Proceedings Of Thirteenth Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 13th Annual GAMTE Proceedings Front Matter:

• Officers of GAMTE
• Reviewers
• Copyright & Licensing Terms
• Purposes and Goals of GAMTE
• Conference Schedule
• Dedication
• Table of Contents

An Alternative Approach To The Traditional Internship, Basil M. Conway, David Erikson, Christopher Parrish, Marilyn Strutchens, Jennifer Whitfield

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

This paper reports the benefits and challenges of incorporating a paired-placement model at four different post-secondary teacher preparation programs in secondary mathematics education. The paired-placement model places two secondary mathematics clinical teachers with one mentor (or cooperating) teacher during their internship experience. Benefits exhibited were increased collaboration, more knowledgeable cooperating teachers, increased sense of community, teaming, pedagogical risk-taking, increased reflective practice, established natural professional learning communities, Plan-Do-Study-Act Cycle (PDSA), and increased accountability. Challenges found through the PDSA cycle include personnel issues, number of days teaching, perceived classroom management preparation, preparing university supervisors, mentors, and teacher candidates, and support for collaboration …