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A metric space (X,d) is declared to be natural if (X,d) determines an up to isomorphism unique group structure (X,+) on the set X such that all the group translations and group inversion are isometries. A group is called natural if it emerges like this from a natural metric. A simple graph X is declared to be natural if (X,d) with geodesic metric d is natural. We look here at some examples and some general statements like that the graphical regular representations of a finite group is always aarXiv:2205.14097v1 fatcat:6fmpzkmuhzfrbcy4n466jw2iem