## Thesis Draft 5

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

Erica Gilliland
# a portfolio of all things mathematics

Posts Tagged ‘ graph ’

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

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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.

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.

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).

In researching the applications of graph theory, Dr. Sharma asked me to consider the graphs of isomers of Carbon-Hydrogen compounds. We derived the formula for possible molecules. If “c” is the number of Carbon atoms and “h” is the number of Hydrogen atoms, then h=2c+2

So if you are given 6 Carbon atoms, it is possible to form a C6H14 molecule. With these fixed number of atoms, there are several possible structures of the graphs. These graphs are called isomers.

In the graphs of Carbon-Hydrogen compounds, each Carbon is connected to 4 other atoms, but each Hydrogen can only connect to one atom. Therefore, since the graphs of molecules must be connected, Hydrogen atoms only connect to Carbon atoms. So the true underlying structure of Carbon-Hydrogen compounds comes from the construction of the Carbon atoms.

Therefore, drawing all non-isomorphic trees on 6 vertices, where the highest degree of any vertex is 4, creates the underlying structure of all possible C6H14 isomers. Below are pictures of the possible isomer constructions. The true construction of each isomer can be formed by attaching a Hydrogen atom to each Carbon atom until the degree of each Carbon atom is 4.

I’m doing some research on LCF notation to see how it relates mathematics to chemistry. LCF notation was originally developed by Joshua Lederberg as a way to represent cyclic graphs in chemistry. It was further developed by Coxeter and Frucht, who dubbed the notation LCF. Here is the original paper written by Lederberg that first uses the notation.

After meeting with my adviser, he suggested some changes for my terminology section and added some new sections. I have now also included personal examples of cubic graphs that I have drawn and LCF notation.

I am a student researcher studying cubic graphs, a peer tutor developing my career skills, and a writer for personal enjoyment and all things mathematics.

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