A quasi-quintic graph is a quasi-regular graph, i.e., a graph such that the degree of every vertex is the same 
except for a single vertex whose degree is 
(Bozóki et al. 2022), where 
. By the handshaking
lemma, such a graph necessarily has odd order: if it has 
vertices, the sum of its vertex
degrees is 
and must be even, so 
is odd.
For 
,
9, 11, ..., the numbers of quasi-quintic graphs are 1, 20, 3749, 1877695, ... (OEIS
A398640). The corresponding numbers of connected
quasi-quintic graphs are 1, 20, 3749, 1877694, ... (OEIS A398641).
The 23 disconnected graphs of order 15 are independently accounted for by the component recurrence 
,
where 
counts connected quasi-quintic graphs on 
vertices and 
counts quintic graphs
on 
vertices. Examples are summarized in the following table and illustrated above.
 | examples |
| 7 | Túran graph  |
| 9 | Harary graph  |
