A connected graph is said to be highly irregular if the neighbors of each vertex have distinct vertex degrees. Highly irregular graphs exist on all orders except 3, 5 and 7, with the numbers of such graphs on 1, 2, ... nodes given by 1, 1, 0, 1, 0, 1, 0, 3, 3, 13, 21, 110, 474, 2545, ... (OEIS A217246).
Small examples are illustrated above with the vertex degree of each node labeled and summarized in the following table.
| vertex count | highly irregular graphs |
| 1 | singleton graph |
| 2 | path
graph  |
| 4 | path
graph  |
| 6 | A graph |
| 8 | 4-centipede graph |
Larger examples on 
and 9 nodes are illustrated above, again with the vertex
degree of each node labeled.
Highly irregular graphs are apex, bridges, and nonhamiltonian.
If 
is a vertex with maximum vertex degree 
in a highly irregular graph, then 
is adjacent to exactly one vertex of
degree 1, 2, ..., 
(Alavi et al. 2022).
The maximum vertex degree in a highly irregular graph on 
vertices is at most 
(Alavi et al. 2022).
