Graph topologies induced by edge lengths

Abstract

Let $G$ be a graph each edge $e$ of which is given a length $\ell (e)$. This naturally induces a distance $d_{\ell }(x,y)$ between any two vertices $x,y$, and we let $\ell -TOP$ denote the completion of the corresponding metric space. It turns out that several well studied topologies on infinite graphs are special cases of $\ell -TOP$. Moreover, it seems that $\ell -TOP$ is the right setting for studying various problems. The aim of this paper is to introduce $\ell -TOP$, providing basic facts, motivating examples and open problems, and indicate possible applications.

Final version, to appear in special issue (Infinite graphs, Banff workshop) of Discrete Mathematics

References in corpus (1)

• Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

Cited by in corpus (1)

• Geodetic topological cycles in locally finite graphs