Graph topologies induced by edge lengths

Let G be a graph each edge e of which is given a length l(e). This naturally induces a distance d(l)(x, y) between any two vertices x. y, and we let vertical bar G vertical bar(l) denote the completion of the corresponding metric space. It turns out that several well-studied topologies on infinite graphs are special cases of vertical bar G vertical bar(l). Moreover, it seems that vertical bar G vertical bar(l) is the right setting for studying various problems. The aim of this paper is to introduce vertical bar G vertical bar(l), providing basic facts, motivating examples and open problems, and indicate possible applications. Parts of this work suggest interactions between graph theory and other fields, including algebraic topology and geometric group theory. (C) 2011 Published by Elsevier B.V.