Year: 2012
Type: preprint
Abstract: We show that the series of all walks between any two vertices of any (possibly weighted) directed graph $\mathcal{G}$ is given by a universal continued fraction of finite depth and breadth involving the simple paths and simple cycles of $\mathcal{G}$. A simple path is a walk forbidden to visit any vertex more than once. We obtain an explicit formula giving this continued fraction. Our results are based on an equivalent to the fundamental theorem of arithmetic: we demonstrate that arbitrary walks on $\mathcal{G}$ factorize uniquely into nesting products of simple paths and simple cycles, where nesting is a product operation between walks that we define. We show that the simple paths and simple cycles are the prime elements of the set of all walks on $\mathcal{G}$ equipped with the nesting product. We give an algorithm producing the prime factorization of individual walks, and obtain a recursive formula producing the prime factorization of sets of walks. Our results have already found applications in machine learning, matrix computations and quantum mechanics.
Source: arXiv (Cornell University)
Authors: Giscard, P. -L., Thwaite, S. J., Jaksch, D.
Institution: University of Oxford
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2198744598