Introduction

The earliest origins of graph theory can be found in puzzles and game, including Euler's Königsberg 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 topology. 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 Poincaré (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 Oswald Veblen [1880–1960] shows how Poincaré 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.