Archive for November, 2011

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)

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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).

Developing a Code for Isomers

During our meeting this week, Dr. Sharma asked me to look into developing a code for constructing isomers. Prüfer code was developed as a way to numerically describe the construction of labelled graphs. Since isomers are not labeled graphs (hydrogen atoms are not identified separately, and neither are carbon atoms), a Prüfer code would not apply directly to isomers. So instead, we discussed dividing the atoms of carbon-hydrogen molecules into classes.

For example, consider the following isomer of a C6H14 molecule:

All of the hydrogen atoms would be put into Class 1. Next we are left with the underlying structure of the carbon atoms. We said that the central carbon atom that is connected to four other carbon atoms is the strongest. So Class 2 would consist of all the carbon atoms furthest from this central, strong atom (which in this example, is only one…the bottom carbon).

Removing this atom from our picture, we are left with the central, strong carbon atom and four connecting carbon atoms. So Class 3 would consist of those four connecting atoms, and the last class, Class 4, would consist of that central, strong carbon atom.

The number of atoms in each class thus generates a code: 14-1-4-1

Our hope was that this code would give us unique constructions of isomers when assembled in the reverse order. In our example, we would know to start with one carbon atom. Then connect four carbon atoms to it. Next, since the placement of the sixth carbon atom is not important, connect one atom to any of the last four. And last, connect 14 hydrogen atoms to complete the molecule.

Unfortunately, our code did not prove to be unique or simple. Consider constructing a C6H14 molecule from the following code: 14-2-3-1. Starting with 1 carbon atom, you attach 3 at a distance 1 from this atom. Then you attach 2 at a distance 2 from the beginning atom. And last, you attach the 14 hydrogen atoms to complete the molecule. Following these steps, you can end up with two nonisomorphic constructions of C6H14 (illustrated below).

The problem develops because generating the code for the bottom isomer would be difficult. There are two carbon atoms that are attached to 3 other carbon atoms, so their “strength” is the same. Thus, the code produced from this molecule would be 14-4-2, which is inconsistent with the code used to build the molecule. Therefore, the code we wanted to develop to express the unique isomers of carbon-hydrogen molecules is faulty.

Isomers of C6H14

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.

Gray Code & Graph 64

This week, we studied Graph 64 from The Foster Census and applied a gray code to it’s vertices.

Since Graph 64 is based on a Hamilton cycle, we next compared each vertex to the third vertex (besides it’s two neighbors in the Hamilton cycle) that it is connected to. Our goal was to see if there is a pattern that exists between the intersection and differences of the two gray codes. Unfortunately, we did not find a correlation.

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