Denote by G = (V, ∼) a graph which V is the vertex set and ∼ is an adjacency relation on a subset of V × V. In this paper, the good distance graph is defined. Let (V, ∼) and (V′, ∼′) be two good distance graphs, and φ : V → V′ be a map. The following theorem is proved: φ is a graph isomorphism ⇔ φ is a bounded distance preserving surjective map in both directions ⇔ φ is a distance k preserving surjective map in both directions (where k < diam (G) / 2 is a positive integer), etc. Let D be a division ring with an involution over...
