More about this week and my project

June 10, 2009

So let’s see, what has happened in the last few days? My group has basically been working on making a big catalog. I think I need to explain a little more about my project. I will do my best to explain it so you at least get an idea. We start with a graph, a simple graph (one that does not have any loops or double edges) and we make a matrix from it. There are lots of different types of matrices you can make. One matrix we made for each graph was an adjacency matrix.
(I couldn't get the graph to paste here so use the link to look at it)
http://en.wikipedia.org/wiki/File:6n-graf.svg

For a graph, each of the dots is called a vertex and the line connecting the dots is called an edge. For an adjacency matrix, if there is an edge connecting two vertices, you put a 1 in the matrix, if there is no edge connecting the two vertices, you put a zero. Below is the matrix for this graph.

0 1 0 0 1 0
1 0 1 0 1 0
0 1 0 1 0 0
0 0 1 0 1 1
1 1 0 1 1 0
0 0 0 1 0 0

I’ll explain the first row. There is a one in the 2nd column and the 5th column because vertex one is connected to vertex 2 and vertex 5 by an edge.
One of my group members made a computer program to make the adjacency matrix so we only had to input the edges and the program made the matrix for us. Once we have the matrix, we used a math software called Maple to find the eigenvalues of the matrix. I’m not sure how to explain an eigenvalue. Once we have the eigenvalues, we take the absolute value of each and add them up which gives us the energy. Does that help?

So the last few days we have been calculating the eigenvalues and energy for the adjacency, Laplacian and distance matrices for 208 graphs, all the graphs with 1 to 6 vertices. We have a catalog that pictures all these graphs and all the graphs with 7 vertices but there are a TON of graphs with 7 vertices, over 1000, so we are starting with just the graphs with 6 or fewer vertices and eventually we hope to catalog the graphs with 7 vertices as well. Once we have all the data we will be looking for patterns, making conjectures and hopefully proving them! We will be looking at things like what happens to the energy if you add/delete an edge? It is taken longer than I expected to finish the catalog/database for the first 208 graph but I think we will finish tomorrow!

Other than that, I’ve been watching TV/movies, reading (just finished “Three Cups of Tea”), knitting, hanging out and getting to know everyone better.