Resources for Teaching Discrete Mathematics: Early Writings on Graph Theory: Topological Connections

The earliest origins of graph theory can be found in puzzles and game, including Euler’s Konigsberg Bridge Problem and Hamilton’s Icosian Game. A second important branch of mathematics that grew out of these same humble beginnings was the study of position (“analysis situs”), known today as topology1. In this project, we examine some important connections between algebra, topology and graph theory that were recognized during the years from 1845 1930. The origin of these connections lie in work done by physicist Gustav Robert Kirchhoff [1824 1887] on the flow of electricity in a network of wires. Kirchhoff showed how the current flow around a network (which may be thought of as a graph) leads to a set of linear equations, one for each circuit in the graph. Because these equations are not necessarily independent, the question of how to determine a complete set of mutually independent equations naturally arose. Following Kirchhoff’s publication of his answer to this question in 1847, mathematicians slowly began to apply his mathematical techniques to problems in topology. The work done by the French mathematician Henri Poincare [1854 1912] was especially important, and laid the foundations of a new subject now known as “algebraic topology.” This project is based on excerpts from a 1922 paper in which the American mathematician Oswald Veblen [1880 1960] shows how Poincare formalized the ideas of Kirchhoff. An American mathematician born in Iowa, Veblen’s father was also a mathematician who taught mathematics and physics at the State University of Iowa. At that time, graduate programs in mathematics were relatively young in the United States. A member of the first generation of American mathematicians to complete their advanced work in the United States rather than Europe, Oswald Veblen completed his Ph.D. at the University of Chicago in 1903. He remained in Chicago for two years before joining the mathematics faculty at Princeton. In 1930, he became the first faculty member of the newly founded Institute for Advanced Study at Princeton.2 A talented fund-raiser and organizer, Veblen also served on the Institute’s Board of Trustees in its early years.