A uniquely graceful graph is a graph that has exactly one fundamentally distinct graceful labeling. Uniquely graceful graphs
for 
,
..., 7, are illustrated above.
There exists a uniquely graceful graph on 12 vertices (the fourth of the smallest graceful octic graphs in the numbering of GraphData) which is also an identity graph, meaning it has a unique graceful labeling even among permutation of labels (though of course not counting subtractive complementation). Such graphs might be termed identically uniquely graceful.
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 connected
graphs and uniquely graceful simple graphs with
no isolated vertices 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 vertices |
| A379573 | 1, 1, 2, 4, 1, 4, 2, 19, ... | simple connected graphs |
