Archive for the ‘ Honors Thesis ’ Category

Enumerating Unlabeled Trees

At the Nebraska Conference for Undergraduate Women in Mathematics (big related announcement to come soon) in January, I presented part of my thesis work. The following abstract summarizes what I talked about. The presentation was mostly just for visuals, so you will have to wait until my final thesis is published to read more & fill in some of the blanks.

Abstract:

One of the earliest papers on graph theory was written by Euler in 1736. Since then, graphs have been studied by the likes of Foster, Coxeter, Biggs, and Peterson. Applications extend beyond the world of pure mathematics into the areas of engineering, computer programming, chemistry, and more. Yet, after three hundred years of intense study, there are many problems that are open or in need of improvements. In this talk, we will examine an improved algorithm that we developed for counting the number of unlabeled trees where the degree of each vertex is less than or equal to four, which translates into counting the number of isomers of alkanes.

Enumerating Unlabeled Trees (pdf of presentation slides)

Enumerating Unlabeled Trees

I attended the Nebraska Conference for Undergraduate Women in Mathematics this year and gave a 15 minute talk on the improved algorithm I have developed as part of my thesis. I will be giving the talk again at the Butler Undergraduate Research Conference in April of this year after my final thesis is published.

Enumerating Unlabeled Trees (pdf of slides)

Thesis Update

I haven’t posted anything about my thesis in a while, and since it’s set to be finished in about a month, I’m going to wait until it’s published, then post the final paper.

Thesis Draft 5

From here on, each draft of my thesis will be constructed using PCTeX.

Thesis Draft 5

Thesis Draft 4 (PCTeX)

Dr. Sharma asked me to rewrite what I had so far for my thesis using PCTeX. PCTeX is a software that uses TeX, a mathematical language, to produce documents. I have been learning the software and language as I go, so these drafts are truly works in progress.

Thesis Draft 4 (PCTeX)

Thesis Draft 3

This draft reflects the new organization of my thesis and main topics we aim to cover. Some of the topics have already been discussed, but I did not have time to add them to this draft of the paper. See blog posts for up-to-date information on what topics I have already researched.

Thesis Draft 3

Isomers of Alkanes

After studying C6H14, we became interested in the structure of carbon hydrogen compounds in general. I discovered, later, that molecules that are made up of only carbon and hydrogen atoms that contain no cycles are called alkanes. Isomers are compounds that have the same number of carbon and hydrogen atoms, but have different structures. Since hydrogen atoms do not add to the basic structure of alkanes, it is sufficient to study the underlying structure of the carbon atoms.

In connection to graph theory, studying the structure of the carbon atoms in alkane isomers is equivalent to studying the structure of nonisomorphic trees with no vertex of degree greater than four. The following graphs (representing trees, or the structure of carbon atoms in alkanes) are grouped vertically by the number of vertices.

The result that we began to notice while comparing the groups of isomers was that the beginning of a Fibonacci sequence appeared: 1, 1, 2, 3, 5.

Seeing that this pattern was developing, we tried to think of a reason why. But once I tested C7H16, the pattern fell apart. Instead of finding 8 isomers, as a Fibonacci sequence would produce, there are 9 (see below).

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