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 | G | ℓ  denote the completion of the corresponding metric space. It turns out that several well-studied topologies on infinite graphs are special cases of | G | ℓ . Moreover, it seems that | G | ℓ is the right setting for studying various problems. The aim of this paper is to introduce | G | ℓ , providing basic facts, motivating examples and open problems, and indicate possible applications. of this work suggest interactions between graph theory and other fields, including algebraic topology and geometric group theory.