Topological Graph Theory Dan Archdeacon Summary: Topological Graph Theory A Survey Dan Archdeacon Dept. of Math. and Stat. University of Vermont Burlington, VT, USA 05405 e­mail: dan.archdeacon@uvm.edu 1 Introduction Graphs can be represented in many different ways: by lists of edges, by incidence relations, by adjacency matrices, and by other similar structures. These representations are well suited to computer algorithms. Historically, however, graphs are geometric objects. The vertices are points in space and the edges are line segments joining select pairs of these points. For example, the points may be the vertices and edges of a polyhedron. Or they may be the intersections and traffic routes of a map. More recently, they can represent computer processors and communication channels. These pictures of graphs are visually appealing and can convey structural information easily. They reflect graph theory's childhood in ``the slums of topology.'' Topological graph theory deals with ways to represent the geometric real­ ization of graphs. Typically, this involves starting with a graph and depicting Collections: Mathematics