Theoretical mathematics success
took hard work, not magic © March 13, 2009, Norwich University Office of Communications

A smiling Jeremy Holden gestures as he explains vertex magic labeling in front of a black board filled with equations.

photo by Jay Ericson, staff Jeremy Holden’s work on a mathematics paper about vertex-magic labeling may have helped him secure a competitive summer internship.

Jeremy Holden, a third-year mathematics student at Norwich University, opened a pad of paper and drew a simple triangle; three points, or “vertices,” connected by lines called “edges.” It’s not geometry, he explained. The points could represent anything, and the lines aren’t necessarily straight.

“Now, let’s play a game,” he said, using his pen to ascribe a number to each vertex and edge in the drawing. He used the numbers 1 through 6 in no particular order without skipping or repeating. To complete the puzzle, Holden positioned numerals so that adding numbers at a vertex and both of its intersecting edges created a sum equal to that of either of the other vertices added to its associated edges. When the numbers add up, in this case to 12, you’ve achieved vertex-magic labeling.

Two figures from a nine-figure graph that demonstrate vertex-magic labeling.

Two figures from a nine-figure graph that demonstrate vertex-magic labeling. The full puzzle uses numbers 1 to 58. For all figures in the graph, the sum of the three numbers at the confluence of two edges is 103.

It’s a simple puzzle, but grows to astonishing complexity as the number of vertices, edges and figures within the graph increase. Vertex-magic labeling lies at the heart of a proof that Holden, of Westminster, Vt., and Norwich mathematics Professor Dan McQuillan published in Discrete Mathematics, a peer-review journal for research in fields such as graph and enumeration theory, and combinatorics. Using research grants, Holden spent two summers and much of the school years working through complicated problems to help McQuillan find graph labelings other mathematicians theorized were nonexistent.

Computer power, he said, offered little assistance. His progress happened in a quiet space with pen, paper and time to think.

“It’s a creative pursuit,” said Holden. “It’s about solving problems. It’s not about number crunching.”

Inadvertently illustrating his point, Holden drew a more complicated graph with four shapes. Counting up the total of numerals needed to label all vertices and edges, he wrote down 24. After a moment, he crossed it out. The correct number was 26.

He proved that he was very eager to ask the right kind of questions. He’s willing to work very hard, and he doesn’t give up easily.

~ Prof. Dan McQuillan

He’s been ill, Holden explained, and not really at the top of his game. One misconception about mathematicians, he believes, is that they have a flawless grasp on number computation. He considers himself more a theoretician with a love of puzzles and the patience to work through complex problems.

“You have to keep coming up with ideas and trying, trying, trying,” said Holden. “Most of the ideas you try will be wrong.”

McQuillan explained that the range of permutations you find in a labeling problem that uses just a few numbers or figures quickly becomes more than a computer is useful in analyzing. Mathematicians must search for general formulas that work for infinite varieties of graphs, then use computers for testing. He and Holden would come up with ideas and send them to his brother, a computer science professor at Western Illinois University, to build a testing program.

“My feeling was that if a computer program couldn’t give me an answer in one day, then I was asking it the wrong questions,” he said.

McQuillan, who has been working on vertex-magic labeling problems for years, met Holden in Norwich’s Putnam Math Club, where a group of students meet weekly to wrestle with interesting problems outside of a mathematics curriculum. In Holden, he recognized creativity and determination.

“He proved that he was very eager to ask the right kind of questions,” said McQuillan. “He’s willing to work very hard, and he doesn’t give up easily.”

The December 2008 paper was vetted and published quickly, McQuillan said. He believes it presents new, interesting methodologies and questions. He called the early reviews excellent.

“I think people will read it, and I think it will open doors,” McQuillan said.

Some may have already opened for Holden. After publication, he was accepted for a paid internship at the Center for Discrete Mathematics & Theoretical Computer Science at Rutgers University, where he will continue to work on graph problems. He’s very excited about the honor, Holden said. While he can’t say whether the published paper brought the opportunity, he imagines it didn’t hurt.

“Having that under my belt certainly made me a competitive applicant,” he said.

Cathy Frey, dean of mathematics and sciences at Norwich, called both the paper and internship “amazing” honors for an undergraduate.

“It’s an exceptional thing,” she said, referring to the internship. “No other student that I know of from Norwich has had that opportunity.”

Moving from graph to graph in his notebook, Holden explained the meaning of complex ideas in the simple language of a natural teacher. That’s what he’d like to pursue, and hopes to earn his doctorate in mathematics to work with motivated students at the college level.

“I’d love to teach,” said Holden. “It’s fun to see people understand something they didn’t before.”