Two-point resistances of symmetric polyhedral networks

Abstract

A random walk along the edges of a polyhedron consists of starting at a particular vertex, choosing one of its adjacent neighbors at random, stepping to that neighbor, and then repeating the process until a specified vertex is reached. The average number of steps it takes to get from a given vertex to another is called the hitting time. For polyhedra with some degree of symmetry, there are groups of vertices that have the same hitting time from a given starting vertex. These groups form symmetrically equivalent layers within the polyhedron. This paper explores a number of novel approaches to calculating these layers based on Euclidean distance, minimum length paths, and the number of edges of each face of the polyhedron. After the hitting times have been found, the equivalent resistance between any two vertices of the network is easily calculated. The physical relevance of this problem is discussed, a comparison with other approaches to solving it is made and some open questions are mentioned.