A quasi-cubic graph is a quasi-regular graph, i.e., a finite simple
graph such that the vertex degree of every vertex is the same 
except for a single vertex
whose vertex degree is 
(Szádoczki et al. 2022), with 
. By the handshaking
lemma, such a graph necessarily has odd graph order: if it has 
vertices, the sum
of its vertex degrees is 
and must be even, so 
is odd.
For 
, 7, 9, ..., the numbers of quasi-cubic graphs are
1, 4, 28, 214, 1999, 22186, ... (OEIS A398638).
The corresponding numbers of connected quasi-cubic
graphs are 1, 4, 27, 208, 1958, 21879, ... (OEIS A398639).
Of the quasi-cubic graphs on 19 vertices, 28526 are
disconnected. The sole disconnected quasi-cubic
graph through nine vertices is the graph
disjoint union 
, whose components
are the wheel graph 
and the tetrahedral graph 
. Examples are illustrated above and are summarized
in the table below.
 | quasi-cubic graphs |
| 5 | wheel graph  |
| 7 | Harary graph  , Moser spindle |
| 9 | Harary
graph  ,  -unit-distance-forbidden graph,  |
