A uniquely graceful graph is a graph that has exactly one fundamentally distinct graceful labeling. Uniquely graceful graphs
for 
,
..., 7, are illustrated above.
The numbers of uniquely graceful graphs of various types on 
, 2, ... vertices are summarized in the following table.
Knuth (2024, Problem 97) enumerates uniquely graceful simple graphs in the connected
and no isolated point cases up to 8 vertices.
| OEIS | counts | type |
| A379574 | 1, 1, 2, 5, 2, 5, 11, 33, ... | simple graphs |
| A379572 | 0, 1, 2, 4, 1, 5, 10, 29, ... | simple graphs with no isolated points |
| A379573 | 1, 1, 2, 4, 1, 4, 2, 19, ... | simple connected graphs |
