How to create a metric in a connected locally finite abstract graph?

Question

First of all sorry for my English. I want to create a metric in a graph $(V, E)$ where $V$ is the countable set of vertices, $E$ the set of edges. The graph is non-directed, connected, locally finite. The purpose of the metric is to define an isometry where the edges are isometric to $[0,1]$ or to $\mathbb{S}^1$. My first idea was define the metric in $V$ (the distance between two vertices would be the minimum length of the paths between them), but I have to give a metric in $E$ because I have to make the edges isometric to some sets. Also, i have to make the edges a set and not a point, can I make the edges some type of segment between the vertices?

Answer 1

Maybe you might find this article useful: https://arxiv.org/pdf/1912.07549v2.pdf