### Summer Combo in Vermont - July 13th 2012

Speaker List

If you are interested in contributing a talk please send email to Jo Ellis-Monaghan (jellis-monaghan@smcvt.edu) with the author(s), title, and abstract for the talk by July 1st.

This list of speakers below, will be updated as talks are submitted.

**Dan Archdeacon, The University of Vermont**

**Title: **How to draw many triple systems at once

**
Abstract:** A Steiner Triple System (STS) is a set of triples collectively containing any pair exactly once. A nice way to depict an STS is by drawing each triple as a triangle on some surface. We explore how to draw many different STS's simultaneously.

**Jo Ellis-Monaghan, Saint Michael's College**

**Title:** Graph theory tools for DNA origami assembly

**Astract: **Recently, ‘origami folding’ methods of DNA self-assembly have emerged, wherein a single scaffolding strand of DNA traces the construct exactly once, and then short helper strands bond to this strand to fold and staple it into the desired configuration. While originally applied to solid 2D structures, and later solid 3D structures, that are ‘filled’ by the strands of DNA, this technique may also be adapted to graphical structures such as polyhedral skeletons.

For origami constructions, questions focus on finding a ‘threading’ of the graph or multigraph, that is, a walk for the scaffolding strand through the graph, ideally covering each edge exactly once. This may be just an Eulerian circuit if the graph happens to be Eulerian. However, for non-Eulerian graphs, the problem of adapting either the graph or the circuit to enable the construction raises new mathematical questions. There is a topology-based existence theorem for threadings of arbitrary graphs, but it (necessarily) allows undesirable double-backs. Obviously, if the construction can be adapted by adding augmenting edges to make the graph Eulerian, a threading can easily be found. However, the problem becomes more challenging with the added constraints that the embedding of the graph in 3-space is often fixed ahead of time, and that, to the extent possible, the threading should follow faces and hence not cross itself at any vertex.

We describe the application, give examples of using graph theoretical techniques to establish optimal threadings for some 3D graphical structures, and share some threading results.

**Dan MacQuillan, Norwich University**

**Title: **Vertex-magic graphs

**Abstract:** A total labeling of a graph G=(V,E) with v vertices and e edges, is a bijective map from the integers 1,2,3,… v+e onto V union E. The weight of a vertex is the sum of its label added to the sum of all of its incident edge labels. The labeling is called vertex-magic with magic constant h if the weight of every vertex is h.

One major goal is to determine which graphs have a vertex-magic total labeling. Another major goal is to understand which integers can be magic constants. We will present selective results dealing with both problems. An interesting conjecture is MacDougall’s conjecture, which posits that every regular graph of degree at least 2 (with one exception) has a vertex-magic total labeling. More generally, we ask about the extent to which the graph structure, as opposed to the degree sequence, will determine the existence of such a labeling. The work is very accessible and there remain many open problems.

**Chris McDaniel, Endicott College**

**Title:** 1-Skeleta

**Abstract: **1-skeleta are "linearized" regular graphs that were invented in the 1990's by mathematicians interested in the topology of torus actions on smooth manifolds. I will discuss various aspects of these beautiful structures from a combinatorial point of view.

**Iain Moffatt, University of South Alabama**.

**Title:** Permanents and the Jones polynomial

**Abstract**: The permanent of a square matrix is defined in a similar way to the determinant, but without using signs. It is well-known that permanents count the cycle covers of a digraph. I will use this combinatorial interpretation of the permanent to express the Jones polynomial (which is a knot invariant) as the permanent of a matrix. I will then discuss some computational implications of this result.

This talk is based on joint work with Martin Loebl.

**Sam Northshield, SUNY-Plattsburgh**

**Title: ** On the Fibonacci shift

**Abstract: ** Every integer can be represented as a sum of distinct Fibonacci numbers. Although such a representation is generally not unique, the replacement of each Fibonacci number by the next one in such a representation actually makes a well-defined function S(n) on the positive integers; the Fibonacci shift.

S(n) is a Beatty sequence: S(n)=[nR+1/R] where R is the golden ratio.

The complementary Beatty sequence is T(n):=[nR+n-1].

The sequence B(n) defined by B(S(n))=B(n), B(T(n))=B(n)+B(n-1) is an analogue of Stern’s sequence and shares many of its properties. Combinatorially, B(n+1) counts the number of representations of n as a sum of distinct Fibonacci numbers.

If k/F(n) increases to x then B(k)/B(F(n)+k) converges to a value, say, Z(x). The function Z(x) is singular and is an analogue of Conway’s box function (and inverse of Minkowski’s ?-function). It can be represented as a sum of powers of 1/R related to the continued fraction representation of x.

Knuth introduced “Fibonacci multiplication” and showed that it is associative. We use S(n) to give a short proof of this fact.

We create the “Fibadic integers” based on a valuation defined by S(n).