Characterization of 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 $(V(G),d_G)$ are characterized up to isometry for all finite connected $G$ by David C. Kay and Gary Chartrand in 1964. The main result of the paper expands this characterization on the infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding into $(V(G),d_G)$ for suitable $G$.