Characterization of the geodesic distance on infinite graphs

Let G be a connected graph and let d_G be the geodesic distance on V (G). The metric spaces
(V (G), d_G) were characterized up to isometry for all finite connected G by David C. Kay and Gary
Chartrand in 1965. The main result of this paper expands this characterization on innite connected
graphs. We also prove that every metric space with integer distances between its points admits an
isometric embedding in (V (G), d_G) for suitable G.