Song '13 Charts Graph Theory - Hamilton College
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# Song '13 Charts Graph Theory

The mathematical field of graph theory is a study of just that:  a study of mathematical figures consisting of points and lines connected to them. Yonghyun Song ’13 is serving as an intern for Associate Professor of Mathematics Sally Cockburn on a graph theory project. His work was supported by the Monica Odening Student Internship and Research Fund in Mathematics through the Career Center.

Song’s research involves calculating how many ways one can draw the graph K_2n on a flat plane. K_2n is a figure that consists of two white dots and n black dots. Dots of the same color are not connected, and dots of different colors are all connected. Professor Cockburn started the research in 2010, and she found out algebraic methods to approach this problem. Song is helping Cockburn continue her research from last summer.

Brainstorming for ideas and proofs usually doesn't require much more than paper and pencil, and so much of the research is done without the help of the computer. However, because the numbers involved in the project are too big to compute by hand, computer programs help verify ideas. Though Song frequently works independently, the project is in many ways a collaborative effort between himself and Cockburn. The two meet on Mondays and Thursdays and keep in touch through email about new ideas. Recently, their progress has led them to take a fresh approach to the research, and they have found previous work on the subject that has potential to help them.

On August 5  Song gave a talk titled "Geo-isomorphism classes of K(2,n)"  at MathFest 2011 , the annual summer meeting of the Mathematical Association of America, in Lexington, Ky., based on his summer research with Cockburn.  The project extends work that establishing a connection between the different straight-line drawings of the complete bipartite graph and permutations.  The research question was: how many different permutations give rise to the same straight-line drawing?  To answer the question, a new connection was found between the straight-line drawings and permutation graphs.

Song, a mathematics major who would like to study applied mathematics in graduate school, wanted to  work on the project so he could obtain experience in math research. This particular subject interested him, and he has learned a lot from his work. Song says, “Professor Cockburn and I are both fully devoted to this research, and I receive help from her in every possible way.” The research takes a lot of time and effort, and Song has found that it takes up most of his time on campus this summer. When he has free time, he enjoys spending it with friends and watching movies.

Yonghyun Song is a graduate of Korean Minjok Leadership Academy in Hoengseong, Gangwon, South Korea.

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