Archive for September, 2011

Elevator Speech

It is great to meet you, my name is Erica Gilliland, and I am a junior Mathematics and Actuarial Science major at Butler University. I’m looking for a position that will allow me to use my research and communication skills. Over the past few years, I have been strengthening these skills through my work at Information Commons, where we provide students, staff and faculty with research assistance and academic technology training and support. Eventually, I would like to translate the job skills I have learned into a pure mathematics environment where I can be more specifically involved in research and teaching. I will be attending graduate school after graduation and working towards my doctorate.

Advertisements

LCF Notation for 3-6 Graphs on 14, 16, and 18 Vertices

Tuesday I learned how to express Hamiltonian graphs in LCF notation. LCF notation begins with a Hamiltonian graph. From there, it shows you how to draw a cubic graph by starting at any vertex. Below is the LCF notation for the 3-6 graphs I have drawn on 14, 16, and 18 vertices.

3-6 Cage
LCF Notation [ 5, -5]^7

3-6 Cage

 3-6 Graph on 16 Vertices
LCF Notation [5, -5, 6, -5, 6, -5, 5, 6, -6, 6, -6, -5, 5, -6, 5, -6]

 3-6 Graph on 18 Vertices
LCF Notation [5, 9, -5]^6

Tracing a Hamiltonian Cycle on the 3-6 Graph on 18 Vertices

A few weeks ago I constructed a 3-6 graph on 18 vertices. My strategy then was to start with 3 sets of 6 vertices, two of which were already connected to form a Hamilton cycle of length 6. My object then was to interconnect the vertices to create a 3-6 graph, and I was able to do this successfully.

 Graph A

My adviser then wanted me to rearrange the vertices to see if there existed a Hamiltonian graph of length 18 in the above 3-6 graph I had drawn. I was able to successfully rearrange the vertices to clearly illustrate that there was one (image below).

Graph B

He also asked me to write an algorithm so that starting at any vertex, I could trace the Hamilton cycle of length 18. The algorithm is included below.

Algorithm:

Name the following three groups of vertices in Graph A.
Outer Vertices {C, I, L, D, R, O}
Middle Vertices {F, H, K, N, Q, A}
Inner Vertices {B, G, J, E, P, M}

Choose any vertex in Graph A

You can start anywhere in the pattern and it loops until you have returned to your starting vertex. The following two rules must also be kept:

1) Visit each vertex once and only once

2) The first time you travel Middle-Middle or Outer-Outer sets the direction you will travel throughout the cycle (clockwise or counterclockwise). Once set, this direction must always be followed.

Pattern:
Outer
Inner
Middle
Middle
Inner
Outer

3-6 Graph on 16 Vertices

Last week I was challenged to draw a 3-g Graph on 16 vertices. I managed to draw a 3-5 graph, but after talking with my advisor, realized that because 3-6 cage is on 14 vertices, there existed a 3-6 graph on 16 vertices. So below is my construction of a cubic graph on 16 vertices with the maximum girth (6).

3-6 Cage (corrected)

Tonight, as I was pulling apart my 3-6 cage and attempting to make it into a Hamiltonian cubic graph, I realized it was in fact not a 3-6 cage. A cycle of length 5 appeared that had not been obvious to me before (highlighted below).

5-3 graph on 14 vertices

After I had constructed what I thought was a 6-3 Cage, I looked it up online to compare the common construction to mine. Because of that, I already know how to construct the 6-3 Cage, so I included it below.

6-3 Cage

Proof for Finite Abelian Groups

Theorem:
All finite abelian groups have the property that the product of all their elements is e, except when there is one and only one element of order two. In that case, the product of all the elements is that unique element of order two.

Proof:

Resizing Images using Microsoft Office PowerPoint

After I created the Microsoft Office Picture Manager Quick Guide, my advisor showed me how to resize images from within PowerPoint. Since the professor we created the guides for wants to resize images in PowerPoint, we thought this would be the best option. Some professors have Microsoft Office 2010, while others are still on 2007, so I developed a guide for both versions.

Compressing Images using PowerPoint 2007 (PDF)

Compressing Images using PowerPoint 2010 (PDF)

%d bloggers like this: