Graph topologies induced by edge lengths

Abstract: Let be a graph each edge of which is given a length . This naturally induces a distance between any two vertices , and we let  denote the completion of the corresponding metric space. It turns out that several well-studied topologies on infinite graphs are special cases of . Moreover, it seems that is the right setting for studying various problems. The aim of this paper is to introduce , 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. [Copyright &y& Elsevier]