  We introduce the notion of Levenshtein graphs, an analog to Hamming graphs
but using the edit distance instead of the Hamming distance; in particular,
Levenshtein graphs allow for underlying strings (nodes) of different lengths.
We characterize various properties of these graphs, including a necessary and
sufficient condition for their geodesic distance to be identical to the edit
distance, their automorphism group and determining number, and an upper bound
on their metric dimension. Regarding the latter, we construct a resolving set
composed of two-run strings and an algorithm that computes the edit distance
between a string of length $k$ and any single-run or two-run string in $O(k)$
operations.
